Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

TL;DR

This paper introduces a systematic method to generate all non-isomorphic topological representations of complete and bipartite graphs within beyond-planarity classes, leading to new bounds on their properties.

Contribution

The paper presents a novel technique for generating all non-isomorphic representations of certain graphs under beyond-planarity constraints, surpassing traditional combinatorial methods.

Findings

Generated all non-isomorphic topological representations for selected graph classes.

Achieved new tight bounds for maximum edge density in beyond-planarity classes.

Demonstrated the technique's effectiveness across various graph classes.

Abstract

Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size of the largest complete or complete bipartite graph belonging to it. Overcoming the limitations of standard combinatorial arguments, we present a technique to systematically generate all non-isomorphic topological representations of complete and complete bipartite graphs, taking into account the constraints of the specific class. As a proof of concept, we apply our technique to various beyond-planarity classes and achieve new tight bounds for the aforementioned parameter.

Tables

Table 1. Table 1 : Known results and our findings. For each class, we present the largest complete and complete bipartite graphs that belong to this class (col. “ ∈ \in ”), and the smallest ones that do not (col. “ ∉ \notin ”). Color gray indicates weaker results that follow from other entries.

		complete				complete bipartite
Class	Density	$\in$	Ref.	$\notin$	Ref.	$\in$	Ref.	$\notin$	Ref.
1-planar	$4n-8$	$K_6$	[20, Fig.1]	$K_7$	[39, Thm.1]	$K_{3,6}$	[20, Fig.2]	$K_{3,7}$	[20, Lem.4.2]
						$K_{4,4}$	[20, Fig.3]	$K_{4,5}$	[20, Lem.4.3]
2-planar	$5n-10$	$K_7$	[13, Fig.7]	$K_8$	Char.4.1	$K_{3,10}$	[6, Lem.1]	$K_{3,11}$	[6, Lem.1]
						$K_{4,6}$	Char.4.2	$K_{4,7}$	Char.4.2
						$K_{4,5}$		$K_{5,5}$	Char.4.2 [34]
3-planar	$\frac{11}{2}n-11$	$K_8$	Char.4.1	$K_9$	Char.4.1	$K_{3,14}$	[6, Lem.1]	$K_{3,15}$	[6, Lem.1]
						$K_{4,9}$	Char.4.3	$K_{4,10}$	Char.4.3
						$K_{5,6}$	Char.4.3	$K_{5,7}$	Char.4.3
						$K_{5,6}$		$K_{6,6}$	Char.4.3
4-planar	$6n-12$	$K_9$	Char.4.1	$K_{10}$	Char.4.1	$K_{3,18}$	[6, Lem.1]	$K_{3,19}$	[6, Lem.1]
						$K_{4,11}$	Obs.4.4	$K_{4,19}$
						$K_{5,8}$	Obs.4.4	$K_{5,19}$
						$K_{6,6}$	Obs.4.4	$K_{6,19}$
fan-planar	$5n-10$	$K_7$	[13, Fig.7]	$K_8$	Char.4.5	$K_{4,n}$	[33, Fig.3]	$K_{5,5}$	Char.4.6
fan-crossing	$4n-8$	$K_6$	[20, Fig.1]	$K_7$	[19, Thm.1]	$K_{3,6}$		$K_{3,7}$	Char.4.8
free						$K_{4,6}$	Char.4.8	$K_{4,7}$
						$K_{4,5}$		$K_{5,5}$	Char.4.8
gap-planar	$5n-10$	$K_8$	[11, Fig.7]	$K_9$	[11, Thm.23]	$K_{3,12}$	[11, Fig.7]	$K_{3,14}$	[10, Thm.1]
						$K_{4,8}$	[11, Fig.9]	$K_{4,9}$	Obs.4.10
						$K_{5,6}$	[11, Fig.9]	$K_{5,7}$	[11]
						$K_{5,6}$		$K_{6,6}$	[10, Thm.1]
RAC	$4n-10$	$K_5$	[25, Fig.5]	$K_6$	[22, Thm.1]	$K_{3,4}$	[21, Fig.4]	$K_{3,5}$	[21, Thm.2]
						$K_{3,4}$		$K_{4,4}$	[21, Thm.2]
quasiplanar	$\frac{13}{2}n-20$	$K_{10}$	[14, Fig.1]	$K_{11}$	[3, Thm.5]	$K_{4,n}$	[33, Fig.3]	–
						$K_{5,18}$	Obs.4.12	?
						$K_{6,10}$	Obs.4.12	?
						$K_{7,7}$	Obs.4.12	$K_{7,52}$	[3, Thm.5]

Table 2. Table 2 : A comparison of the number of drawings reported by our algorithm with the elimination of isomorphic drawings (col. “Non-Iso”) and without it (col. “All”) for the classes of 1 1 1 - and 2 2 2 -planar graphs; the corresponding execution times (in sec.) to compute these drawings are reported next to them.

	complete					complete bipartite
Class	Graph	Non-Iso.	Time	All	Time	Graph	Non-Iso.	Time	All	Time
1-planar	$K_4$	2	0.043	8	0.043	$K_{2,3}$	3	0.061	34	0.061
	$K_5$	1	0.043	30	0.206	$K_{3,3}$	2	0.049	84	0.539
	$K_6$	1	0.020	120	0.737	$K_{3,4}$	3	0.065	960	5.642
	$K_7$	0	0.006	0	0.448	$K_{4,4}$	2	0.044	1584	10.871
						$K_{4,5}$	0	0.010	0	7.198
	total:	4	0.112	158	1.434	total:	10	0.229	2662	24.311
2-planar	$K_4$	2	0.028	8	0.028	$K_{2,3}$	6	0.090	76	0.090
	$K_5$	4	0.105	294	2.661	$K_{3,3}$	19	0.254	2352	10.571
	$K_6$	6	0.233	2664	3.292	$K_{3,4}$	71	1.458	52248	244.964
	$K_7$	2	0.119	8400	55.323	$K_{4,4}$	38	1.152	168624	1128.457
	$K_8$	0	0.029	0	51.321	$K_{4,5}$	37	1.826	1200384	8135.843
						$K_{5,5}$	0	0.357	0	12639.293
	total:	14	0.514	11366	112.625	total:	171	5.137	1423684	22159.218

Table 3. Table 3 : A summary of the required time (in sec.) and of the number of general and non-isomorphic drawings for different complete and complete bipartite graphs.

	complete				complete bipartite
Class	Graph	General	Non-Iso.	Time	Graph	General	Non-Iso.	Time
1-planar	$K_4$	8	2	0.043	$K_{2,3}$	34	3	0.061
	$K_5$	13	1	0.043	$K_{3,3}$	14	2	0.049
	$K_6$	4	1	0.020	$K_{3,4}$	16	3	0.065
	$K_7$	0	0	0.006	$K_{4,4}$	5	2	0.044
					$K_{4,5}$	0	0	0.010
	total:	25	4	0.112	total:	69	10	0.229
2-planar	$K_4$	8	2	0.028	$K_{2,3}$	76	6	0.090
	$K_5$	89	4	0.105	$K_{3,3}$	243	19	0.254
	$K_6$	56	6	0.233	$K_{3,4}$	526	71	1.458
	$K_7$	38	2	0.119	$K_{4,4}$	310	38	1.152
	$K_8$	0	0	0.029	$K_{4,5}$	318	37	1.826
					$K_{5,5}$	0	0	0.357
	total:	191	14	0.514	total:	1473	171	5.137
3-planar	$K_4$	8	2	0.042	$K_{2,3}$	76	6	0.234
	$K_5$	109	5	0.195	$K_{3,3}$	678	69	1.802
	$K_6$	548	39	0.953	$K_{3,4}$	7141	1188	16.969
	$K_7$	648	39	3.459	$K_{4,4}$	24058	2704	97.801
	$K_8$	20	3	1.153	$K_{4,5}$	44822	7653	310.194
	$K_9$	0	0	0.065	$K_{5,5}$	20043	1899	199.908
					$K_{5,6}$	2516	438	47.396
					$K_{6,6}$	0	0	4.822
	total:	1333	88	5.867	total:	99334	13957	679.126
4-planar	$K_4$	8	2	0.040	$K_{2,3}$	76	6	0.108
	$K_5$	109	5	0.222	$K_{3,3}$	968	102	2.146
	$K_6$	1374	95	4.080	$K_{3,4}$	32454	6194	163.000
	$K_7$	14728	1266	79.842	$K_{4,4}$	681196	81817	34096.183
	$K_8$	7922	833	84.725	$K_{4,5}$	?	?	?
	$K_9$	353	35	33.672
	$K_{10}$	0	0	1.175
	total:	24494	2236	203.756	total:	?	?	?
5-planar	$K_4$	8	2	0.059
	$K_5$	109	5	0.259
	$K_6$	1752	119	4.716
	$K_7$	83710	8318	1396.781
	$K_8$	1190765	138750	262419.413
	$K_9$	285847	29939	32299.196
	$K_{10}$	0	0	2783.813
	total:	1562191	177133	298904.237

Table 4. (b) Part B: Results concerning the remaining graph classes considered in this paper.

	complete				complete bipartite
Class	Graph	General	Non-Iso.	Time	Graph	General	Non-Iso.	Time
fan-planar	$K_4$	8	2	0.034	$K_{2,3}$	76	6	0.110
	$K_5$	89	5	0.133	$K_{3,3}$	127	9	0.292
	$K_6$	147	39	0.226	$K_{3,4}$	295	43	0.757
	$K_7$	75	39	0.405	$K_{4,4}$	255	29	0.972
	$K_8$	0	0	0.196	$K_{4,5}$	324	48	1.624
					$K_{5,5}$	0	0	0.637
	total:	319	22	0.994	total:	1077	135	4.392
fan-crossing	$K_4$	8	2	0.049	$K_{2,3}$	34	3	0.057
free	$K_5$	13	1	0.054	$K_{3,3}$	38	5	0.092
	$K_6$	4	1	0.038	$K_{3,4}$	28	5	0.098
	$K_7$	0	0	0.009	$K_{4,4}$	19	4	0.106
					$K_{4,5}$	16	2	0.075
					$K_{5,5}$	0	0	0.012
	total:	25	4	0.150	total:	135	19	0.440
gap-planar	$K_4$	14	2	0.135	$K_{2,3}$	169	14	0.256
	$K_5$	243	10	0.366	$K_{3,3}$	1425	266	4.359
	$K_6$	739	237	4.726	$K_{3,4}$	16898	7466	170.396
	$K_7$	1124	665	13.943	$K_{3,5}$	148527	56843	12032.226
	$K_8$	1	1	16.347	$K_{4,5}$	199778	148367	28457.751
	$K_9$	0	0	0.019	$K_{4,6}$	408476	246318	132622.664
					$K_{4,7}$	173271	101428	32958.628
					$K_{4,8}$	5981	4015	2708.278
					$K_{4,9}$	0	0	99.583
	total:	2121	915	35.536	total:	954525	564717	209054.141
quasiplanar	$K_4$	8	2	0.082	$K_{2,3}$	76	6	0.187
	$K_5$	109	5	0.193	$K_{3,3}$	604	53	0.859
	$K_6$	936	63	1.820	$K_{3,4}$	11902	2248	34.073
	$K_7$	16505	1607	69.943	$K_{4,4}$	386241	46711	11328.401
	$K_8$	173199	20980	4044.264	$K_{4,5}$	?	?	?
	$K_9$	209248	23011	35163.772
	$K_{10}$	81	9	7593.865
	$K_{11}$	0	0	5.225
	total:	400086	45677	46879.164	total:	?	?	?

Full text

Overcoming the limitations of standard combinatorial arguments, we present a technique to systematically generate all non-isomorphic topological representations of complete and complete bipartite graphs, taking into account the constraints of the specific class. As a proof of concept, we apply our technique to various beyond-planarity classes and achieve new tight bounds for the aforementioned parameter.

Generation of topological representations

1 Introduction

Beyond-planarity is an active research area concerned with combinatorial properties of non-planar graphs that lie in the “neighborhood” of planar graphs. More concretely, these graphs allow for non-planar drawings in which certain geometric or topological crossing configurations are forbidden. The most studied beyond-planarity classes, with early results dating back to 60’s [9, 42], are the $k$-planar graphs [39], which forbid an edge to be crossed more than $k$ times, and the $k$-quasiplanar graphs [4], which forbid $k$ mutually crossing edges; see Figs. 1(a)-1(b).

More recently, several other classes have been suggested (e.g., [6, 11]), also motivated by cognitive experiments [32, 37] indicating that the absence of certain types of crossings helps in improving the readability of a drawing; for a survey, refer to [24]. Some of the most studied are:

(i) fan-planargraphs, in which no edge can be crossed by two independent edges or by two adjacent edges from different directions [12, 13, 33],

(ii) fan-crossing freegraphs, in which no edge can be crossed by two adjacent edges [16, 19],

(iii) gap-planargraphs, in which each crossing is assigned to one of its two involved edges, such that each edge can be assigned at most one crossing [11], and

(iv) RAC graphs, in which edge crossings occur only at right angles [22, 23, 25]; see Figs. 1(c)-1(e).

Note that all the aforementioned graph classes are topological, i.e., each edge is represented as a simple curve, with the only exception of the class of RAC graphs, which is a purely geometric graph class, i.e., each edge must be represented as a straight-line segment. In this work, we refer to the aforementioned topological graph classes as beyond-planarity classes of topological graphs.

A common characteristic of these graph classes is that their edge density is at most linear in the number of vertices, e.g., $1$-planar graphs with $n$ vertices have at most $4n-8$ edges [39]; see Table 1. Another common measure to determine the extent of a specific class is the size of the largest complete or complete bipartite graph belonging to it [11, 14, 20, 21], which also provides a lower bound on their chromatic number [30] and has been studied in related fields (e.g., [8, 17, 26, 31]).

For $1$-planar graphs, Czap and Hudák [20] proved that the complete graph $K_{n}$ is $1$-planar if and only if $n\leq 6$, and that the complete bipartite graph $K_{a,b}$, with $a\leq b$, is $1$-planar if and only if $a\leq 2$, or $a=3$ and $b\leq 6$, or $a=b=4$. An analogous characterization is known for the class of RAC graphs by Didimo et al. [21, 22], who proved that $K_{n}$ is a RAC graph if and only if $n\leq 5$, while $K_{a,b}$, with $a\leq b$, is a RAC graph if and only if $a\leq 2$, or $a=3$ and $b\leq 4$. For the classes of $3$-quasiplanar (also known as quasiplanar), gap-planar, and fan-crossing free graphs, characterizations exist only for complete graphs, i.e., $K_{n}$ is quasiplanar if and only if $n\leq 10$ [3, 14], gap-planar if and only if $n\leq 8$ [11], and fan-crossing free if and only if $n\leq 6$ [19, 20]; Table 1 gives more details.

To prove the “if part” of these characterizations, one has to provide a certificate drawing of the respective graph. The proof for the “only if part” is generally more complex, as it requires arguments to show that no such drawing exists.

One of the main techniques is provided by the linear edge density of the graph classes; e.g., $K_{7}$ is neither $1$-planar nor fan-crossing free, as it has more than $4n-8$ edges [19, 39]. However, this technique has a limited applicability; e.g., for $2$-planar and fan-planar graphs, which have at most $5n-10$ edges, it only ensures that $K_{9}$ is not a member of these classes. Proving that $K_{8}$ is also not a member requires a different approach. The limitations are even more evident for complete bipartite graphs, as they are sparser than the complete ones (see Section 4).

Another technique consists of showing that the minimum number of crossings required by any drawing of a certain graph (as derived by, e.g., the Crossing Lemma [2, 5, 38] or closed formulas [43]) exceeds the maximum number of crossings allowed in the considered graph class. However, this technique only applies to classes that impose such restrictions, e.g., gap- and $1$-planar graphs [10, 20].

This difficulty in finding combinatorial arguments to prove that certain complete (bipartite) graphs do not belong to specific classes often results in the need of a large case analysis on the different topological representations of the graph. Beside the proofs in [21, 34], we give in Appendix 0.E another example of a combinatorial proof that, based on a tedious case analysis, yields a characterization of the complete bipartite fan-crossing free graphs. The range of the cases in these proofs justifies the need of a tailored approach to systematically explore them.

Our contribution. We suggest a technique to engineer the analysis of all topological representations of a graph that satisfy certain beyond-planarity constraints. Our technique does not extend to classes of geometric graphs, and is tailored for complete and complete bipartite graphs, as we exploit their symmetry to reduce the search space, by discarding equivalent topological representations.

In Section 3, we present an algorithm to generate all possible representations of such graphs under different topological constraints on the crossing configurations. Our algorithm builds on two key ingredients, which allow to drastically reduce the search space. First, the representations are constructed by adding a vertex at a time, directly taking into account the topological constraints, thus avoiding constructing unnecessary representations. Second, at each intermediate step, the produced drawings are efficiently tested for equivalence (up to a relabeling of the vertices), which usually allows to discard a large set of them. Using this algorithm, we derived characterizations for several classes, as described in Section 4; Table 1 positions our results with respect to the state of the art. We give preliminary definitions in Section 2 and discuss future directions in Section 5.

2 Preliminaries

We assume familiarity with standard definitions on planar graphs and drawings (see Appendix 0.A). We assume simple drawings, in which there are no self-crossing edges, two edges cross at most once, and adjacent edges do not cross; note this assumption is not without loss of generality [3]. Given a planarization $\Gamma$ of a graph $G$, a half-pathway for a vertex $u$ in $\Gamma$ is a path in the dual of $\Gamma$ from a face incident to $u$ to some face in $\Gamma$, called its destination; see Fig. 2(a). The length of a half-pathway is the number of edges in this path. A half-pathway for $u$ is valid with respect to a beyond-planarity class $\mathcal{C}$ of topological graphs, if $\Gamma$ can be augmented such that

(i) a vertex $v$ is placed in its destination,

(ii) edge $(u,v)$ is drawn as a curve from $u$ to $v$ that crosses only the edges that are dual to the edges in this half-pathway, in the same order, and

(iii) the drawing of $(u,v)$ violates neither the simplicity of the resulting drawing nor the crossing restrictions of class $\mathcal{C}$.

Accordingly, a pathway for an edge $(u,v)$ is a half-pathway for vertex $u$ in $\Gamma$, whose destination is a face incident to vertex $v$. A valid pathway is defined analogously, with the only exception that $v$ is already part of $\Gamma$.

Another ingredient of our algorithm is an equivalence-relationship between different drawings of a graph $G$, i.e, drawings $D_{1}$ and $D_{2}$ of $G$ are isomorphic [35] if there exists a homeomorphism of the sphere transforming $D_{1}$ into $D_{2}$; see Fig. 5 in Appendix 0.A. Namely, $D_{1}$ and $D_{2}$ are isomorphic if $D_{1}$ can be transformed into $D_{2}$ by relabeling vertices, edges, and faces of $D_{1}$, and by moving vertices and edges of $D_{1}$, so that at no time of this process new crossings are introduced, existing crossings are eliminated, or the order of the crossings along an edge is modified. We define a valid bijective mapping between vertices, crossings, edges, and faces of the planarizations $\Gamma_{1}$ and $\Gamma_{2}$ of $D_{1}$ and $D_{2}$ such that:

(P.1) if an edge $(v_{1},w_{1})$ is mapped to an edge $(v_{2},w_{2})$ in $\Gamma_{1}$ and $\Gamma_{2}$, respectively, and $v_{1}$ is mapped to $v_{2}$, then $w_{1}$ is mapped to $w_{2}$;

(P.2) if a face $f_{1}$ is mapped to a face $f_{2}$ in $\Gamma_{1}$ and $\Gamma_{2}$, respectively, and an edge $e_{1}$ incident to $f_{1}$ is mapped to an edge $e_{2}$ incident to $f_{2}$, then the predecessor (successor) of $e_{1}$ is mapped to the predecessor (successor) of $e_{2}$ when walking along the boundaries of $f_{1}$ and $f_{2}$ in clockwise direction. Also, the face incident to the other side of $e_{1}$ is mapped to the face incident to the other side of $e_{2}$.

Clearly, Properties P.1 and P.2 are sufficient for $D_{1}$ and $D_{2}$ to be isomorphic. We believe they are also necessary, but this is beyond the scope of this work. Note that Property P.2 guarantees that two vertices are mapped to each other only if they have the same degree.

Several works [1, 29, 41] that generate simple drawings of complete graphs adopt a weaker definition of isomorphism; two drawings $D_{1}$ and $D_{2}$ are weakly isomorphic [35], if there exists an incidence preserving bijection between their vertices and edges, such that two edges cross in $D_{1}$ if and only if they do in $D_{2}$. Weakly isomorphic drawings that are non-isomorphic differ in the order in which their edges cross [28]. Two simple drawings of a complete graph with the same cyclic order of the edges around each vertex (called rotation system) are weakly isomorphic, and vice versa [28, 40]; hence, generating all simple drawings of a complete graph reduces to finding all rotation systems that determine simple drawings [36]. However, this property holds only for complete graphs [1], while for the complete bipartite graphs, which are more difficult to handle, only partial results exist in this direction [18]. Thus, we decided not to follow this approach.

3 Generation Procedure

Let $\mathcal{C}$ be a beyond-planarity class of topological graphs and let $G$ be a graph with $n\geq 3$ vertices. Assuming that $G$ is either complete or complete bipartite, we describe in this section an algorithm to generate all non-isomorphic simple drawings of $G$ that are certificates that $G$ belongs to $\mathcal{C}$ (if any). We stress that, if $G$ is neither complete nor complete bipartite, then it is a more involved task to recognize isomorphic drawings [27], and thus to eliminate them, which is a key point in the efficiency of our approach (we provide more details in Section 4).

Our algorithm aims at computing a set $\mathcal{S}$ containing all non-isomorphic simple drawings of $G$. In the base of the recursion, graph $G$ is a cycle of length $3$ or $4$, depending on whether $G$ is the complete graph $K_{3}$ or the complete bipartite graph $K_{2,2}$. In the former case, set $\mathcal{S}$ only contains a planar drawing of $K_{3}$, while in the latter case, $\mathcal{S}$ contains a planar drawing and one with a crossing between two non-adjacent edges. This is because, in both cases, any other drawing is either isomorphic to one of these, or non-simple.

In the recursive step, we consider a vertex $v$ of $G$ and assume that we have recursively computed the set $\mathcal{S}$ for $G\setminus\{v\}$. We may assume w.l.o.g. that $\mathcal{S}\neq\emptyset$, as otherwise $G$ would not belong to $\mathcal{C}$. Then, we consider each drawing of $\mathcal{S}$ and our goal is to report all non-isomorphic simple drawings of $G$ that have it as a subdrawing. In other words, we aim at reporting all non-isomorphic simple drawings that can be derived by all different placements of vertex $v$ and the routing of its incident edges in the drawings of $\mathcal{S}$. To this end, let $\Gamma$ be the planarization of one of the drawings in $\mathcal{S}$, and let $u_{1},\ldots,u_{k}$ be the neighbors of $v$ in $G$, where $k=\deg(v)$. If $G$ is a complete graph, then $k=n-1$; otherwise, $G$ is a complete bipartite graph $K_{a,b}$ with $a+b=n$, and $k=a$ or $k=b$ holds.

We start by computing all possible valid half-pathways for $u_{1}$ in $\Gamma$ with respect to $\mathcal{C}$, which corresponds to constructing all possible drawings of edge $(v,u_{1})$ that respect simplicity and the restrictions of class $\mathcal{C}$. To compute these half-pathways, we again use recursion. For each half-pathway, we maintain a list of so-called prohibited edges, which are not allowed to be crossed when inserting edge $(u_{1},v)$, as otherwise either the simplicity or the crossing restrictions of class $\mathcal{C}$ would be violated; see Fig. 2, and Fig. 6 in Appendix 0.B. This list is initialized with all edges incident to $u_{1}$ and is updated at every recursive step.

In the base of this inner recursion, we determine all valid half-pathways for $u_{1}$ of length zero; this means that, for each face $f$ incident to $u_{1}$, we create a half-pathway that starts at $f$ and has its destination also at $f$, which corresponds to placing $v$ in $f$ and drawing edge $(v,u_{1})$ crossing-free. Assume now that we have computed all valid half-pathways of some length $i\geq 0$ in $\Gamma$. We show how to compute all valid half-pathways for $u_{1}$ of length $i+1$ (if any). Consider a half-pathway $p$ of length $i$. Let $f_{p}$ be its destination. Every non-prohibited edge $e$ of $f_{p}$ implies a new half-pathway of length $i+1$, composed of $p$ followed by the edge that is dual to $e$ in $\Gamma$. Note that this process will eventually terminate, since the length of a half-pathway is bounded by the number of edges of $\Gamma$.

For each valid half-pathway $p$ computed by the procedure above, we obtain a new drawing by inserting $(u_{1},v)$ into $\Gamma$ following $p$ and by inserting $v$ into the destination of $p$. It remains to insert the remaining edges incident to $v$, i.e., $(v,u_{2}),\ldots,(v,u_{k})$, into each of these drawings – again in all possible ways. For this, we proceed mostly as above with one difference. Instead of half-pathways, we search for valid pathways for each edge $(v,u_{i})$, $2\leq i\leq k$, i.e., we only consider pathways that start in a face incident to $v$ and end in a face incident to $u_{i}$.

If we find an edge $(v,u_{i})$ for which no valid pathway exists, we declare that $\Gamma$ cannot be extended to a simple drawing of $G$ that respects the crossing restrictions of $\mathcal{C}$. Otherwise, the computed drawings of $G$ are added to $\mathcal{S}$, once all the drawings of $G\setminus\{v\}$ have been removed from it. To maintain our initial invariant, however, once a new drawing is to be added to $\mathcal{S}$, it will be first checked for isomorphism against all previously added drawings. If there is an isomorphic one, then the current drawing is discarded; otherwise, it is added to $\mathcal{S}$.

We stress that we test isomorphism using Properties P.1 and P.2 of a valid bijection. Since these properties are sufficient but we do not know whether they are also necessary, set $\mathcal{S}$ might contain some isomorphic drawings. However, our experiments indicate that the vast majority of them will be discarded.

Testing for isomorphism. We describe a procedure to test whether the planarizations $\Gamma_{1}$ and $\Gamma_{2}$ of two drawings of $G$ comply with Properties P.1 and P.2 of a valid bijection. We start by selecting two edges $e_{1}=(v_{1},w_{1})$ and $e_{2}=(v_{2},w_{2})$ in $\Gamma_{1}$ and $\Gamma_{2}$, respectively, whose end-vertices have compatible types (i.e., $v_{1}$ and $v_{2}$ are both real vertices or both crossings, and the same holds for $w_{1}$ and $w_{2}$). We bijectively map $e_{1}$ to $e_{2}$, $v_{1}$ to $v_{2}$, and $w_{1}$ to $w_{2}$, which complies with Property P.1. We call this a base mapping and try to extend it to a valid bijection.

We map to each other the face $f_{1}$ of $\Gamma_{1}$ that is “left” of $e_{1}$ (when walking along $e_{1}$ from $v_{1}$ to $w_{1}$) and the face $f_{2}$ of $\Gamma_{2}$ that is “left” of $e_{2}$ (when walking along $e_{2}$ from $v_{2}$ to $w_{2}$). If the degrees of $f_{1}$ and $f_{2}$ are different, then the base mapping cannot be extended. Otherwise, both $f_{1}$ and $f_{2}$ have degree $\delta$, and we walk simultaneously along their boundaries, starting at $e_{1}$ and $e_{2}$ respectively; in view of Property P.2, for each $i=1,\ldots,\delta$, we bijectively map the $i$-th vertex (either real or crossing) of $f_{1}$ to the $i$-th vertex of $f_{2}$, and the $i$-th edge of $f_{1}$ to the $i$-th edge of $f_{2}$. If a crossing is mapped to a real vertex, or if the degrees of two mapped vertices are different, then the base mapping cannot be extended.

If the vertices and edges of $f_{1}$ and $f_{2}$ have been mapped successfully, we proceed by considering the two maximal connected subdrawings $\Gamma^{\prime}_{1}$ and $\Gamma^{\prime}_{2}$ of $\Gamma_{1}$ and $\Gamma_{2}$, respectively, such that each edge of $\Gamma^{\prime}_{1}$ and $\Gamma^{\prime}_{2}$ has at least one face incident to it that is already mapped. Consider an edge $e^{\prime}_{1}$ of $\Gamma^{\prime}_{1}$ that is incident to only one mapped face $f^{\prime}_{1}$ (such an edge exists, as long as the base mapping has not been completely extended). Let $e^{\prime}_{2}$ be the edge of $\Gamma^{\prime}_{2}$ mapped to $e^{\prime}_{1}$; note that $e^{\prime}_{2}$ must be incident to a face $f^{\prime}_{2}$ that is mapped to $f^{\prime}_{1}$ and to a face that is not mapped yet. We map to each other the faces incident to $e^{\prime}_{1}$ end $e^{\prime}_{2}$ that are not mapped yet, and we proceed by applying the procedure described above (i.e., we walk along the boundaries of $f^{\prime}_{1}$ and $f^{\prime}_{2}$ simultaneously, while ensuring that the mapping remains valid). If this procedure can be performed successfully, then we have computed two subdrawings $\Gamma^{\prime\prime}_{1}$ and $\Gamma^{\prime\prime}_{2}$, such that $\Gamma^{\prime}_{1}\subseteq\Gamma^{\prime\prime}_{1}$, $\Gamma^{\prime}_{2}\subseteq\Gamma^{\prime\prime}_{2}$, and each edge of them has at least one face incident to it that is already mapped. Hence, we can recursively apply the aforementioned procedure to $\Gamma^{\prime\prime}_{1}$ and $\Gamma^{\prime\prime}_{2}$.

Drawings $\Gamma_{1}$ and $\Gamma_{2}$ are isomorphic, if the base mapping can be eventually extended. If not, then we have to consider another base mapping and check whether this can be extended. Note that the case where $e_{1}$ is bijectively mapped to $e_{2}$, $v_{1}$ to $w_{2}$, and $w_{1}$ to $v_{2}$ defines a different base mapping than the one we were currently considering. If none of the base mappings can be extended, then we consider $\Gamma_{1}$ and $\Gamma_{2}$ as non-isomorphic. To reduce the number of base mappings that we have to consider, we first count the number of edges of $\Gamma_{1}$ and $\Gamma_{2}$ whose endpoints are both real vertices, both crossings, and those consisting of one real vertex and one crossing. These numbers have to be the same in $\Gamma_{1}$ and $\Gamma_{2}$. Since it is enough to consider base mappings only restricted to one of the three types of edges, we choose the type with the smallest positive number of occurrences. We summarize the above discussion in the following theorem.

Theorem 3.1

Let $G$ be a complete (or a complete bipartite) graph and let $\mathcal{C}$ be a beyond-planarity class of topological graphs. Then, $G$ belongs to $\mathcal{C}$ if and only if, under the restrictions of class $\mathcal{C}$, our algorithm returns a valid drawing of $G$.

4 Proof of Concept - Applications

In this section we use the algorithm described in Section 3 to test whether certain complete or complete bipartite graphs belong to specific beyond-planarity graph classes. We give corresponding characterizations and discuss how our findings are positioned within the literature. Our lower bound examples are drawings that certify membership to particular beyond-planarity graph classes, computed by an implementation (https://github.com/beyond-planarity/complete-graphs) of our algorithm; for typesetting reasons we redrew them. Our upper bounds are the smallest corresponding instances reported as negative by our algorithm.

The class of k-planar graphs.

We start our discussion with the case of complete graphs. As already mentioned in the introduction, the complete graph $K_{n}$ is $1$-planar if and only if $n\leq 6$ [20].

For the case of complete $2$-planar graphs, the fact that a $2$-planar graph with $n$ vertices has at most $5n-10$ edges [39] implies that $K_{9}$ is not a member of this class. Fig. 7 in [13], on the other hand, shows that $K_{7}$ is $2$-planar. We close this gap by showing, with our implementation, that even $K_{8}$ is not $2$-planar.

For the cases of complete $3$-, $4$-, and $5$-planar graphs, the application of a similar density argument as above proves that $K_{10}$, $K_{11}$, and $K_{19}$ are not $3$-, $4$-, and $5$-planar, respectively [2, 38]. With our implementation, we could show that even $K_{9}$ is not $3$-planar, while $K_{10}$ is neither $4$- nor $5$-planar. On the other hand, our algorithm was able to construct $3$- and $4$-planar drawings of $K_{8}$ and $K_{9}$, respectively; see Figs. 3(a) and 3(b). Note that a $6$-planar drawing of $K_{10}$ can be easily derived from the $4$-planar drawing of $K_{9}$ in Fig. 3(b) by adding one extra vertex inside the red colored triangle. We have the following characterization.

Characterization 4.1

For $k\in\{1,2,3,4\}$, the complete graph $K_{n}$ is $k$-planar if and only if $n\leq 5+k$. Also, $K_{n}$ is $5$-planar if and only if $n\leq 9$.

Note that the $3$-planarity of $K_{8}$ implies that the chromatic number of $3$-planar graphs is lower bounded by $8$. Analogous implications can be derived for the classes of $4$-, $5$-, and $6$-planar graphs. Another observation that came out from our experiments is that, up to isomorphism, $K_{6}$ has a unique $1$-planar drawing, $K_{7}$ has only two $2$-planar drawings, and $K_{8}$ has only three $3$-planar drawings, while the number of non-isomorphic $4$-planar drawings of $K_{9}$ is significantly larger, namely $35$. For more details, refer to Table 2, and to Table 3(a) in Appendix 0.D.

Consider now a complete bipartite graph $K_{a,b}$ with $a\leq b$. Note that $a\leq 2$ implies that $K_{a,b}$ is planar; thus, it trivially belongs to all beyond-planarity graph classes. Also, recall that $K_{a,b}$ is $1$-planar if and only if $a\leq 2$, or $a=3$ and $b\leq 6$, or $a=b=4$ [20]. Further, a recent combinatorial result states that $K_{3,b}$ is $k$-planar if and only if $b\leq 4k+2$ [6]. So, in the following we assume $a\geq 4$.

For complete bipartite $2$-planar graphs, the fact that a bipartite $2$-planar graph with $n$ vertices has at most $3.5n-7$ edges [7] implies that neither $K_{4,15}$ nor $K_{5,8}$ is $2$-planar. With our implementation, we could show that $K_{4,7}$ and $K_{5,5}$ are not $2$-planar, while $K_{4,6}$ is (see Fig. 3(c)), yielding the following characterization.

Characterization 4.2

*The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is 2-planar if and only if

(i) $a\leq 2$, or

(ii) $a=3$and $b\leq 10$, or

(iii) $a=4$and $b\leq 6$.

As opposed to the corresponding $2$-planar case, there exists no upper bound on the edge density of $3$-planar graphs tailored for the bipartite setting. The upper bound of $5.5n-11$ edges [38] for general $3$-planar graphs with $n$ vertices does not provide any negative instance for $a\leq 5$, and only proves that $K_{6,b}$, with $b\geq 45$, is not $3$-planar. With our implementation, we could provide significant improvements, by showing that $K_{4,10}$, $K_{5,7}$, and $K_{6,6}$ are not $3$-planar, while $K_{4,9}$ and $K_{5,6}$ are (see Figs. 3(d) and 3(e)), which yields the following characterization.

Characterization 4.3

*The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is 3-planar if and only if

(i) $a\leq 2$, or

(ii) $a=3$and $b\leq 14$, or

(iii) $a=4$and $b\leq 9$, or

(iv) $a=5$and $b\leq 6$.

For complete bipartite $4$-planar graphs, we were unable to derive a characterization, but only some partial results, because the search space becomes drastically larger and, as a consequence, our generation technique could not terminate. To give an intuition, note that $K_{4,4}$ has 81817 non-isomorphic $4$-planar drawings, which makes the computation of the corresponding non-isomorphic drawings of $K_{4,5}$ infeasible in reasonable time; see also Appendix 0.D.

However, we were at least able to report some positive certificate drawings by slightly refining our generation technique. Instead of computing all possible non-isomorphic simple drawings of graph $K_{a-1,b}$ or $K_{a,b-1}$, in order to compute the corresponding ones for $K_{a,b}$, we only computed few samples, hoping that we will eventually find a positive certificate drawing. With this so-called DFS-like approach, we managed to derive $4$-planar drawings for $K_{4,11}$, $K_{5,8}$, and $K_{6,6}$; see Fig. 4. We summarize these findings in the following observation.

Observation 4.4

*The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is 4-planar if

(i) $a\leq 2$, or

(ii) $a=3$and $b\leq 18$, or

(iii) $a=4$and $b\leq 11$, or

(iv) $a=5$and $b\leq 8$, or

(v) $a=6$and $b=6$.

Further, $K_{a,b}$ is not 4-planar if $a\geq 3$ and $b\geq 19$.*

The class of fan-planar graphs.

We start our discussion with complete graphs. The fact that a fan-planar graph with $n$ vertices has at most $5n-10$ edges [33] implies that $K_{9}$ is not fan-planar, while Fig.7 in [13] shows that $K_{7}$ is. With our implementation, we showed that $K_{8}$ is not fan-planar, even relaxing the requirement that an edge crossed by two or more adjacent edges must be crossed from the same direction; see, e.g., [15]. This yields the following characterization.

Characterization 4.5

The complete graph $K_{n}$ is fan-planar if and only if $n\leq 7$.

Consider now a complete bipartite graph $K_{a,b}$ with $a\leq b$. For $a\leq 4$, $K_{a,b}$ is fan-planar for any value of $b$ [33]. On the other hand, the fact that a bipartite fan-planar graph has at most $4n-12$ edges [7] implies that $K_{5,9}$ is not fan-planar. Using our implementation, we could show that even $K_{5,5}$ is not fan-planar (again by relaxing the requirement of having the crossings from the same direction). These two results together imply the following characterization.

Characterization 4.6

The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is fan-planar if and only if $a\leq 4$.

The class of fan-crossing free graphs.

A characterization for the case of complete graphs can be derived by combining two known results. First, $K_{6}$ is fan-crossing free, as it is $1$-planar; see Table 3(a) in Appendix 0.D additionally shows that, up to isomorphism, $K_{6}$ has a unique fan-crossing free drawing. Second, the fact that a fan-crossing free graph with $n$ vertices has at most $4n-8$ edges [19] implies that $K_{7}$ is not fan-crossing free. Hence, we have the following characterization.

Characterization 4.7** (Cheong et al. [19], Czap et al. [20])**

*The complete

graph $K_{n}$ is fan-crossing free if and only if $n\leq 6$.*

As already stated, for the complete bipartite fan-crossing free graphs, we provide in Appendix 0.E a combinatorial proof of their characterization. The same result was also obtained by our implementation; see Table 3(a) in Appendix 0.D.

Characterization 4.8

*The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is fan-crossing free if and only if

(i) $a\leq 2$, or

(ii) $a\leq 4$and $b\leq 6$.

The class of gap-planar graphs.

A characterization of the complete gap-planar graphs has already been provided [11] as follows.

Characterization 4.9** (Bae et al. [11])**

The complete graph $K_{n}$ is gap-planar if and only if $n\leq 8$.

For the case of complete bipartite graphs, Bae et al. [11] proved that $K_{3,12}$, $K_{4,8}$, and $K_{5,6}$ are gap-planar, while $K_{3,15}$, $K_{4,11}$, and $K_{5,7}$ are not. These negative results were derived using the technique discussed in Section 1 that compares the crossing number of these graphs with their number of edges, which is an upper bound to the number of crossings allowed in a gap-planar drawing. By refining this technique, Bachmaier et al. [10] proved that even $K_{3,14}$, $K_{4,10}$, and $K_{6,6}$ are not gap-planar. Hence, towards a characterization the cases that are left open are $K_{3,13}$ and $K_{4,9}$. Here, we address one of these two open cases by showing that $K_{4,9}$ is not gap-planar, thus yielding the following observation.

Observation 4.10

*The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is gap-planar if

(i) $a\leq 2$, or

(ii) $a=3$and $b\leq 12$, or

(iii) $a=4$and $b\leq 8$, or

(iv) $a=5$and $b\leq 6$.

Further, $K_{a,b}$ is not gap-planar if

(i) $a=3$and $b\geq 14$, or

(ii) $a=4$and $b\geq 9$, or

(iii) $a=5$and $b\geq 7$, or

(iv) $a\geq 6$and $b\geq 6$.

The class of quasiplanar graphs.

A characterization for the complete quasiplanar graphs can be also derived by combining two known results. Namely, the fact that a quasiplanar graph with $n$ vertices has at most $6.5n-20$ edges [3] implies that $K_{11}$ is not quasiplanar, while $K_{10}$ is in fact quasiplanar [14].

Characterization 4.11** (Ackerman et al. [3], Brandenburg [14])**

The complete graph $K_{n}$ is quasiplanar if and only if $n\leq 10$.

Consider now a complete bipartite graph $K_{a,b}$ with $a\leq b$. First, we observe that for $a\leq 4$, graph $K_{a,b}$ is quasiplanar for any value of $b$, since it is even fan-planar [33]. On the other hand, the fact that a quasiplanar graph with $n$ vertices has at most $6.5n-20$ edges [3] does not provide any negative answer for $a\leq 6$, while for $a=7$ it only implies that $K_{7,52}$ is not quasiplanar. We stress that we were not able to find any improvement on the latter result. The reason is the same as the one that we described for the class of complete bipartite $4$-planar graphs. To give an intuition, we note that $K_{4,4}$ has in total 46711 non-isomorphic quasiplanar drawings, which makes the computation of the corresponding non-isomorphic drawings of $K_{4,5}$ infeasible in reasonable time; see also Appendix 0.D. Notably, using the DFS-like variant of our algorithm, we were able to derive at least positive certificate drawings for $K_{5,18}$, $K_{6,10}$, and $K_{7,7}$; see Figs. 7, 8(a), and 8(b) in Appendix 0.C. We summarize these findings in the following observation.

Observation 4.12

*The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is quasiplanar if

(i) $a\leq 4$, or

(ii) $a=5$and $b\leq 18$, or

(iii) $a=6$and $b\leq 10$, or

(iv) $a=7$and $b\leq 7$.

Further, $K_{a,b}$ is not quasiplanar if $a\geq 7$ and $b\geq 52$.*

5 Conclusions and Open Problems

We conclude this work by noting that our results also have some theoretical implications. In particular, $K_{5,5}$ was conjectured in [7] not to be fan-planar; Characterization 4.6 settles in the positive this conjecture. By Characterization 4.6 and Observation 4.10, we deduce that $K_{5,5}$ is a certificate that there exist graphs which are gap-planar but not fan-planar. Since $K_{4,9}$ is fan-planar but not gap-planar, the two classes are incomparable, which answers a related question posed in [11] about the relationship between 1-gap-planar graphs and fan-planar graphs.

We stress that the elimination of isomorphic drawings is a key step in our algorithm, as shown in Table 2. For example, to test whether $K_{5,5}$ is $2$-planar without the elimination of intermediate isomorphic drawings, one would need to investigate 1423684 drawings, while in the presence of this step only 171. This significantly reduced the required time to roughly 5 seconds, including the time to perform all isomorphism tests and eliminations. We provide further insights in Appendix 0.D, where we broaden our description to the other classes (see Table 3(a)).

Our work leaves two open problems. Is it possible to extend our approach to graphs that are neither complete nor complete bipartite, e.g., to $k$-trees or to $k$-degenerate graphs (for small values of $k$)? A major difficulty is that, in the absence of symmetry, discarding isomorphic drawings becomes more complex. A general observation from our proof of concept is that our approach was of limited applicability on the classes of complete bipartite $k$-planar graphs, for $k>3$, and complete bipartite quasiplanar graphs, for which we could report partial results. So, is it possible to broaden these results by deriving improved upper bounds on the edge densities of these classes tailored for the bipartite setting (see, e.g., [7]).

Appendix

Appendix 0.A Preliminary Notions and Definitions

In this paper, we consider graphs containing neither multi-edges nor self-loops. Let $G=(V,E)$ be a graph. A drawing of $G$ is a topological representation of $G$ in the plane $\mathbb{R}^{2}$ such that each vertex $v\in V$ is mapped to a distinct point $p_{v}$ of the plane, and each edge $(u,v)\in E$ is drawn as a simple Jordan curve connecting $p_{u}$ and $p_{v}$ without passing through any other vertex. Unless otherwise specified, we consider simple drawings, in which any two edges intersect in at most one point, which is either a common endpoint or a proper crossing. Hence, two edges are not allowed to cross twice (or more times) and no two edges incident to the same vertex are allowed to cross.

A drawing without edge crossings is called planar. Accordingly, a graph that admits a planar drawing is called planar. The planarization of a (non-planar) drawing is the planar drawing obtained by replacing each of its crossings with a dummy vertex. The dummy vertices are referred to as crossing vertices, while the remaining ones (that is, the ones of the original drawing) as real vertices. A planar drawing divides the plane into connected regions, called faces; the unbounded one is called outer face. The degree of a face is defined as the number of edges on its boundary, counted with multiplicity. The dual of a planar drawing $\Gamma$ has a node for each face of $\Gamma$ and an arc between two nodes if the corresponding faces of $\Gamma$ share an edge. Finally, two drawings $\Gamma_{1}$ and $\Gamma_{2}$ of a graph $G$ are isomorphic [35], if there exists a homeomorphism of the sphere that transforms $\Gamma_{1}$ into $\Gamma_{2}$ (see Fig. 5 for an illustration), and weakly isomorphic [35] if there exists an incidence preserving bijection between their vertices and their edges, such that two edges of $\Gamma_{1}$ cross if and only if the corresponding edges of $\Gamma_{2}$ cross.

Appendix 0.B A Running Example

In Fig. 6, we give an example for the insertion of a node $v$ into a crossing-free 4-cycle, such that $v$ is connected to two vertices $u_{1}$ and $u_{2}$. The dashed red edge is the newly inserted edge; the blue edges are prohibited; the turquoise edges are the edges that are marked as prohibited while computing the half-pathway of the red edge. Figs 6(a)–6(j) illustrate all possible ways for drawing edge $(v,u_{1})$. Figs 6(k)–6(o) illustrate all possible ways for inserting edge $(v,u_{2})$ into the drawing of Fig. 6(a). Note that among the drawings that contain the edge $(v,u_{2})$ the drawings of Figs. 6(l) and 6(n) are isomorphic, and the same holds for the drawings of Figs. 6(m) and 6(o). Also, all obtained drawings are legal for the topological graph classes defined in the introduction, except for the class of $1$-planar graphs.

Appendix 0.C Omitted Drawings from Section 4

In this section, we provide drawings certifying that certain complete and complete bipartite graphs belongs to specific beyond-planarity graph classes, which were omitted from Section 4 due to space constraints; refer to Figs. 7 and 8.

Appendix 0.D Further insights from our implementation

In this section, we present some insights from the computations that we made in order to check whether certain complete and complete bipartite graphs belong to specific graph classes; for a summary refer to Table 3(a). Our algorithm was implemented in Java (https://github.com/beyond-planarity/complete-graphs) and was executed on a Windows machine with 2 cores at 2.9 GHz and 8 GB RAM.

As described in Section 3, our algorithm constructs all possible drawings of a certain (complete or complete bipartite) graph by adding a single vertex to the non-isomorphic drawings of the subgraph of it without this vertex. Once a new drawing is obtained in this procedure, we compare it for isomorphism against the already computed ones (and possibly discard it). The total number of produced drawings is reported in the column “General”, while the number of the non-isomorphic ones in the column “Non-Iso.”. The reported times are in seconds and correspond to the total time needed for generation and filtering for isomorphism. The bottommost row of each section in the table corresponds to a negative instance, as no drawing satisfying the constraints of the respective graph class could be found. The class of complete bipartite $4$-planar graphs and the one of complete bipartite quasiplanar graphs form exceptions, as for these classes we were not able to report all non-isomorphic drawings of $K_{4,5}$.

As a typical example, we describe in the following one intermediate step in our computations; refer to the gray colored entry of Part A of Table 3(a). Our algorithm for reporting that $K_{6,6}$ is not a $3$-planar graph generated at some intermediate step all $3$-planar drawings of $K_{5,5}$, based on the non-isomorphic drawings of $K_{4,5}$. The algorithm reported in total 20043 drawings (including isomorphic ones), which were reduced to 1899 due to the elimination of isomorphic ones. These two steps together required 199.908 seconds. The obtained drawings were extended (by adding one additional vertex and its five incident edges) to 2516 drawings of $K_{5,6}$, which were reduced to 438 due to the filtering for isomorphism. None of these drawings could be extended to a $3$-planar drawing of $K_{6,6}$.

The class of complete bipartite $4$-planar graphs and the class of complete bipartite quasiplanar graphs show the limitations of our approach. We start our discussion with the former class. As already mentioned in Section 4, for the class of complete bipartite $4$-planar graphs we were able to report only some partial results (and not a complete characterization). The reason is depicted in Part A of Table 3(a). Observe that, in order to determine the 81817 non-isomorphic drawings of $K_{4,4}$, our implementation needed to generate 681196 drawings starting from the 6194 non-isomorphic drawings of $K_{3,4}$. This growth in the number of non-isomorphic drawings and the time needed to generate them (i.e., 34096 sec.) form a clear indication of the reason why our implementation failed to report all corresponding drawings of $K_{4,5}$. Similar observations can be made for the class of quasiplanar graphs; see Part B of Table 3(a).

We conclude this section by making some additional observations. First, it is eye-catching from both parts of Table 3(a) that the number of general and non-isomorphic drawings of the complete graphs are significantly smaller than the corresponding ones for the complete bipartite graphs, which is explained by the fact that the former are very symmetric and denser.

As it is naturally expected, we also observe that both the number of general drawings and the number of non-isomorphic drawings of a $k$-planar graph increases as $k$ increases (at least for values of $k$ in $\{1,2,3,4\}$). In particular, it seems that this increment becomes significantly larger from $3$- to $4$-planar graphs, both in the complete and in the complete bipartite settings.

Comparing fan-planar and fan-crossing free graphs, which are in a sense complementary to each other, we observe significant differences in the number of general and non-isomorphic drawings. In particular, the number of non-isomorphic drawings of fan-crossing free graphs are always single digits.

We finally observe that it is generally not a time-demanding task to conclude that a graph does not belong to a specific class, once all non-isomorphic drawings of its maximal realizable subgraph have been computed. In fact, the bottommost row of every section in Table 3(a) reports times in the order of few seconds at most.

Appendix 0.E A Combinatorial Proof of Characterization 4.8

In this appendix we give a combinatorial proof for Characterization 4.8 of the complete bipartite fan-crossing free graphs.

Characterization 4.8. *The complete bipartite graph $K_{a,b}$ (with $a\leq b$) is fan-crossing free if and only if

(i) $a\leq 2$, or

(ii) $a\leq 4$and $b\leq 6$.

The sufficiency of the two conditions is proved by observing that graph $K_{2,b}$, for any $b$, is planar, and that graph $K_{4,6}$ is fan-crossing free, as shown in Fig. 3(c).

To prove the necessity of the two conditions, we have to show that neither $K_{5,5}$ nor $K_{3,7}$ is fan-crossing free. We will discuss the two cases separately. However, we first observe some properties that are common for the two cases. Recall that we restrict our analysis to simple drawings, in which there are no self-crossing edges, two distinct edges cross at most once, and adjacent edges do not cross. For a complete bipartite graph $K_{a,b}$, we will denote the two partite sets as $V_{1}=\{u_{1},\dots,u_{a}\}$ and $V_{2}=\{w_{1},\dots,w_{b}\}$.

The following lemma is important to bound the number of possible configurations we have to consider later in the two cases.

Lemma 1

Let $\Gamma_{3,5}$ be a fan-crossing free drawing of $K_{3,5}$. There is a $K_{2,2}$-subgraph of $K_{3,5}$ whose edges do not cross each other in $\Gamma_{3,5}$.

Proof

Consider the $K_{2,2}$-subgraph $G$ induced by vertices $u_{1},u_{2},w_{1},w_{2}$. If no two edges of $G$ cross each other in $\Gamma_{3,5}$, then the statement already follows. So, we may assume that two edges, say $(u_{1},w_{2})$ and $(u_{2},w_{1})$, have a crossing.

We first consider the case in which another crossing exists between edges of $G$; since $\Gamma_{3,5}$ is simple and fan-crossing free, this crossing must be between $(u_{1},w_{1})$ and $(u_{2},w_{2})$. However, it is possible to verify in Fig 9(a) and Fig. 9(b) that it is not possible to realize these two crossings without creating a fan-crossing.

Thus, we may assume that there is no crossing between edges of $G$ other than the one involving $(u_{1},w_{2})$ and $(u_{2},w_{1})$, as in Fig. 10(a), where the three regions $R_{1}$, $R_{2}$, and $R_{3}$, in which vertex $w_{3}$ may lie are shown.

First we consider the case in which $w_{3}$ lies in $R_{1}$ (see Fig. 10(b)). The edge $(u_{1},w_{3})$ can neither cross $(u_{1},w_{1})$ nor $(u_{1},w_{2})$, since adjacent edges do not cross; it cannot cross $(u_{2},w_{1})$, as otherwise $(u_{2},w_{1})$ would cross a fan incident to $u_{1}$; finally, $(u_{1},w_{3})$ cannot cross $(u_{2},w_{2})$ without crossing any other edge. Hence, edge $(u_{1},w_{3})$ does not cross any edge of $G$, and the same holds for edge $(u_{2},w_{3})$. Further, $(u_{1},w_{3})$ and $(u_{2},w_{3})$ do not cross each other, since they are adjacent. However, this implies that there exist even two $K_{2,2}$-subgraphs whose edges do not cross each other in $\Gamma_{3,5}$, namely the one induced by $u_{1},u_{2},w_{3},w_{1}$, and the one induced by $u_{1},u_{2},w_{3},w_{2}$.

We now consider the case in which $w_{3}$ lies in region $R_{2}$ (see Fig. 10(c)); note that the case in which $w_{3}$ lies in $R_{3}$ is symmetric. Similar to the previous case, edge $(u_{1},w_{3})$ cannot cross any edge of $G$. Further, the edge $(u_{2},w_{3})$ can neither cross one of the edges $(u_{2},w_{1})$ or $(u_{2},w_{2})$ (as adjacent edges do not cross), nor $(u_{1},w_{2})$ (this would be a fan-crossing). Hence, the only possible crossing of edge $(u_{2},w_{3})$ is with $(u_{1},w_{1})$, as in Fig. 10(c). In this case, the graph induced by $u_{1},u_{2},w_{2},w_{3}$ is a $K_{2,2}$-subgraph whose edges do not cross each other.

Since we considered all cases, the statement of the lemma follows.

By Lemma 1, we assume in the following that in any fan-crossing free drawing of either $K_{3,7}$ or $K_{5,5}$, the $K_{2,2}$-subgraph induced by vertices $u_{1},u_{2},w_{1},w_{2}$ is such that no two of its edges cross each other. We use the planar drawing of this $K_{2,2}$-subgraph as a starting point to construct all possible fan-crossing free drawings of graph $K_{2,5}$, by adding vertices $w_{3}$, $w_{4}$, and $w_{5}$ one at a time.

Adding vertex $w_{3}$. We consider vertex $w_{3}$ and edges $(u_{1},w_{3})$ and $(u_{2},w_{3})$. By our previous observations for fan-crossing free drawings, edge $(u_{1},w_{3})$ is not allowed to cross $(u_{1},w_{1})$, $(u_{1},w_{2})$ and $(u_{2},w_{3})$. However, a crossing of $(u_{1},w_{3})$ with at most one of $(u_{2},w_{1})$ and $(u_{2},w_{2})$ is possible. For edge $(u_{2},w_{3})$, we can observe that this edge is not allowed to cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, and $(u_{1},w_{3})$, but is allowed to cross at most one of edges $(u_{1},w_{1})$ and $(u_{1},w_{2})$. Fig. 11 shows all the possible drawings that can occur when we add the node $w_{3}$.

Note that many configurations in Fig. 11 are topologically equivalent, namely:

• •

Fig.11(a) and Fig.11(j),

• •

Fig.11(b), Fig.11(d), Fig.11(f), Fig.11(h), Fig.11(k), Fig.11(m), Fig.11(n) and Fig.11(p),

• •

Fig.11(c), Fig.11(e), Fig.11(g), Fig.11(i), Fig.11(l) and Fig.11(o).

So there are basically three different configurations that we have to consider, namely those from Figs. 11(a), 11(b) and 11(c).

Adding vertex $w_{4}$. In the next step we use these three configurations to create all drawings with the additional node $w_{4}$ and the edges $(w_{4},u_{1})$ and $(w_{4},u_{2})$.

First, we consider Fig. 11(a). In this drawing, the edge $(u_{1},w_{4})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$ and $(u_{2},w_{4})$, but it may cross at most one of the edges $(u_{2},w_{1})$, $(u_{2},w_{2})$ or $(u_{2},w_{3})$. Similar, the edge $(u_{2},w_{4})$ cannot cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$ and $(u_{1},w_{4})$, but it may cross at most one of the edges $(u_{1},w_{1})$, $(u_{1},w_{2})$ or $(u_{1},w_{3})$. From these conditions we obtain the three topologically different drawings in Fig. 12.

Now we consider Fig. 11(b). In this drawing, the edge $(u_{1},w_{4})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{2},w_{4})$ and $(u_{2},w_{1})$, but it may cross at most one of the two edges $(u_{2},w_{2})$ or $(u_{2},w_{3})$. Further, edge $(u_{2},w_{4})$ is not allowed to cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{1},w_{4})$, and $(u_{1},w_{3})$, but it may cross at most one of the edges $(u_{1},w_{1})$ or $(u_{1},w_{2})$. From these conditions we obtain the seven drawings in Fig. 13(a)–13(g).

In the last step we consider Fig. 11(c). In this drawing, the edge $(u_{1},w_{4})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{2},w_{4})$, $(u_{2},w_{1})$ and $(u_{2},w_{3})$, and so it may only cross $(u_{2},w_{2})$. Further, the edge $(u_{2},w_{4})$ is not allowed to cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{1},w_{4})$, $(u_{1},w_{2})$ and $(u_{1},w_{3})$, and so it may only cross $(u_{1},w_{1})$. From these conditions we obtain only one single drawing, namely the one shown in Fig. 13(h).

We observe that the configurations from the following figures are topologically equivalent:

• •

Fig.12(b) and Fig.13(a),

• •

Fig.12(c), Fig.13(b), Fig.13(e) and Fig.13(h),

• •

Fig.13(c) and Fig.13(f),

• •

Fig.13(d) and Fig.13(g).

So there are basically five different configurations that we have to consider, namely those from Figs. 12(a), 12(b), 12(c), 13(c) and 13(d).

Adding vertex $w_{5}$. In the next step we use these five configurations to create all drawings with the additional node $w_{5}$ and the edges $(w_{5},u_{1})$ and $(w_{5},u_{2})$.

First, we consider Fig. 12(a). In this drawing, the edge $(u_{1},w_{5})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{1},w_{4})$ and $(u_{2},w_{5})$, but it may cross at most one of the edges $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$ or $(u_{2},w_{4})$. Similarly, the edge $(u_{2},w_{5})$ cannot cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{2},w_{4})$ and $(u_{1},w_{5})$, but it may cross at most one of the edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$ or $(u_{1},w_{4})$. From this, we obtain the three topologically different drawings in Fig. 14.

Second, we consider Fig. 12(b). In this drawing, the edge $(u_{1},w_{5})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{1},w_{4})$, $(u_{2},w_{5})$ and $(u_{2},w_{1})$, but it may cross at most one of the edges $(u_{2},w_{2})$, $(u_{2},w_{3})$ or $(u_{2},w_{4})$. Further, the edge $(u_{2},w_{5})$ cannot cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{2},w_{4})$, $(u_{1},w_{5})$ and $(u_{1},w_{4})$, but it may cross at most one of the edges $(u_{1},w_{1})$, $(u_{1},w_{2})$ or $(u_{1},w_{3})$. From this, we obtain the drawings in Fig. 15.

Now we consider the drawing from Fig. 12(c). In this drawing, the edge $(u_{1},w_{5})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{1},w_{4})$, $(u_{2},w_{5})$, $(u_{2},w_{1})$ and $(u_{2},w_{4})$, but it may cross at most one of the edges $(u_{2},w_{2})$ or $(u_{2},w_{3})$. Further, the edge $(u_{2},w_{5})$ cannot cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{2},w_{4})$, $(u_{1},w_{5})$, $(u_{1},w_{4})$ and $(u_{1},w_{2})$, but it may cross at most one of the edges $(u_{1},w_{1})$ or $(u_{1},w_{3})$. From this, we obtain the drawings in Fig. 16.

Next, we consider the drawing from Fig. 13(c). In this drawing, the edge $(u_{1},w_{5})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{1},w_{4})$, $(u_{2},w_{5})$, $(u_{2},w_{1})$ and $(u_{2},w_{2})$, but it may cross at most one of the edges $(u_{2},w_{3})$ or $(u_{2},w_{4})$. Further, the edge $(u_{2},w_{5})$ cannot cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{2},w_{4})$, $(u_{1},w_{5})$, $(u_{1},w_{3})$ and $(u_{1},w_{4})$, but it may cross at most one of the edges $(u_{1},w_{1})$ or $(u_{1},w_{2})$. From this, we obtain the drawings in Fig. 17.

Finally, we consider the drawing from Fig. 13(d). In this drawing, the edge $(u_{1},w_{5})$ is not allowed to cross edges $(u_{1},w_{1})$, $(u_{1},w_{2})$, $(u_{1},w_{3})$, $(u_{1},w_{4})$, $(u_{2},w_{5})$, $(u_{2},w_{1})$ and $(u_{2},w_{2})$, but it may cross at most one of the edges $(u_{2},w_{3})$ or $(u_{2},w_{4})$. Further, the edge $(u_{2},w_{5})$ cannot cross $(u_{2},w_{1})$, $(u_{2},w_{2})$, $(u_{2},w_{3})$, $(u_{2},w_{4})$, $(u_{1},w_{5})$, $(u_{1},w_{3})$ and $(u_{1},w_{4})$, but it may cross at most one of the edges $(u_{1},w_{1})$ or $(u_{1},w_{2})$. From this, we obtain the drawings in Fig. 18.

We conclude again by observing that the configurations from the following figures are topologically equivalent:

• •

Fig. 14(b) and Fig. 15(a),

• •

Fig. 14(c), Fig. 15(g), Fig. 15(h) and Fig. 16(a),

• •

Fig. 15(b), Fig. 15(i) and Fig. 17(b)

• •

Fig. 15(c), Fig. 15(j), Fig. 16(c), Fig. 16(d), Fig. 17(c), Fig. 17(d), Fig. 17(e), Fig. 17(f), Fig. 18(d) and Fig. 18(f).

• •

Fig. 15(c), Fig. 15(f), Fig. 15(k), Fig. 18(a) and Fig. 18(b),

• •

Fig. 15(e), Fig. 15(l) and Fig. 17(a)

• •

Fig. 16(e), Fig. 18(c), Fig. 18(e) and Fig. 18(f)

Note that Fig. 18(f) appears twice in this list. The reason for this is to make it easy to recognize the similarity of these two sets of drawings. In fact, we observe that there are seven different configurations that we have to consider in the future, namely the configurations from Figs. 14(a), 14(b), 14(c), 15(b), 15(c), 15(d), and 15(e).

From the above discussion, it follows that any fan-crossing free drawing of $K_{2,5}$ is topologically equivalent to one of the seven drawings described above, which are reported again in Fig. 19 for the reader’s convenience. We denote these drawings by $\Gamma_{1},\ldots,\Gamma_{7}$.

Thus, in order to prove that neither $K_{3,7}$ nor $K_{5,5}$ is fan-crossing free, we have to show that it is not possible to add the remaining vertices (one to $V_{1}$ and two to $V_{2}$ for $K_{3,7}$, and three to $V_{2}$ for $K_{5,5}$), without violating the fan-crossing free property, to any of the seven drawings of $K_{2,5}$. To make a systematic analysis of these cases, we will make use of the following lemmas.

Lemma 2

Consider the region $R$ bounded by a crossing-free edge $(u_{1},w_{i})$ and by two crossing edges $(u_{1},w_{j})$, $(w_{i},u_{2})$, with $1\leq i,j\leq 5$, in a fan-crossing free drawing of $K_{a,b}$, with $a\geq 3$ and $b\geq 5$ (see Fig. 20(a)). If at least four vertices of $V_{2}$ lie outside $R$, there is no vertex $u_{h}$, with $3\leq h\leq a$, inside $R$.

Proof

If $R$ contained a vertex $u_{h}$, with $3\leq h\leq a$, then at least one of the three edges bounding $R$ would be crossed by at least two of the edges connecting $u_{h}$ to the at least four vertices of $V_{2}$ outside $R$, hence creating a fan-crossing.

Lemma 3

Consider the region $R_{1}$ bounded by two crossing-free edges $(u_{1},w_{k})$ and $(u_{2},w_{k})$, and by two crossing edges $(u_{1},w_{j})$ and $(w_{i},u_{2})$, with $1\leq i,j,k\leq 5$, in a fan-crossing free drawing of $K_{a,b}$, with $a\geq 3$ and $b\geq 5$ (see Fig. 20(b)–20(c)). Also, consider the two regions $R_{2}$ and $R_{3}$ bounded by $(u_{1},w_{i})$, $(u_{1},w_{j})$, and $(u_{2},w_{i})$, and bounded by $(u_{2},w_{i})$, $(u_{2},w_{j})$, and $(u_{1},w_{j})$, respectively.

Then, there is at most one vertex $u_{h}$, with $3\leq h\leq a$, in $R_{1}$. Also, if $u_{h}$ lies in $R$, then there exist at most two vertices of $V_{2}\setminus\{w_{i},w_{j},w_{k}\}$ outside $R_{1}$; one of these vertices lies in $R_{2}$ and the other in $R_{3}$.

Proof

Suppose that a vertex $u_{h}$, with $3\leq h\leq a$, lies in $R$. We first claim that edges $(u_{h},w_{i})$ and $(u_{h},w_{j})$ cross edges $(u_{1},w_{k})$ and $(u_{2},w_{k})$, respectively; see Fig. 20(b). Namely, edge $(u_{h},w_{i})$ cannot cross $(u_{1},w_{j})$, as this edge already crosses $(u_{2},w_{i})$. Also, edge $(u_{h},w_{j})$ cannot cross $(u_{2},w_{i})$, since this edge already crosses $(u_{1},w_{j})$. Finally, if $(u_{h},w_{i})$ crosses $(u_{2},w_{k})$, and $(u_{h},w_{j})$ crosses $(u_{1},w_{k})$, then $(u_{h},w_{i})$ and $(u_{h},w_{j})$ cross each other, which is not allowed. The claim follows.

We now argue that there exists no vertex $u_{z}$, with $3\leq z\neq h\leq a$, in $R_{1}$. In fact, in this case, $(u_{z},w_{i})$ and $(u_{z},w_{j})$ would have to cross edges $(u_{1},w_{k})$ and $(u_{2},w_{k})$, respectively, for the same reasons as above. Hence, both $(u_{1},w_{k})$ and $(u_{2},w_{k})$ would cross fans incident to $w_{i}$ and $w_{j}$, respectively.

We conclude the proof by considering the possible placement of a vertex $w_{x}\in V_{2}\setminus\{w_{i},w_{j},w_{k}\}$ outside $R_{1}$. Refer to Fig. 20(c). If $w_{x}$ lies neither in $R_{2}$ nor in $R_{3}$, then the only possibility to connect it to $u_{h}$ is to cross both edges $(u_{1},w_{j})$ and $(u_{2},w_{i})$; in this case, however, it is not possible to place any other vertex of $V_{2}\setminus\{w_{i},w_{j},w_{k}\}$ outside $R_{1}$, as this would require an additional crossing between one of the edges bounding $R_{1}$ and an edge incident to $u_{h}$, which would create a fan-crossing. On the other hand, if $w_{x}$ lies in $R_{2}$, we can draw $(u_{h},w_{x})$ by crossing only $(u_{1},w_{j})$, which still leaves the option to place an additional vertex $w_{y}\in V_{2}\setminus\{w_{i},w_{j},w_{k},w_{x}\}$ in $R_{3}$, and draw $(u_{h},w_{y})$ by crossing only $(u_{2},w_{i})$. However, once this edge has been drawn, we cannot add any additional vertex of $V_{2}$ outside $R_{1}$, and the statement follows.

We show how to exploit these lemmas to complete the proof of Characterization 4.8.

Appendix 0.F Graph $K_{5,5}$ is not fan-crossing free

We consider the seven drawings $\Gamma_{1},\dots,\Gamma_{7}$ of $K_{2,5}$ and show that none of them can be extended to a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{1}$. In drawing $\Gamma_{1}$ we have five topologically similar regions, denoted by $R_{1},\ldots,R_{5}$ as in Fig. 21.

We assume w.l.o.g. that $u_{3}$ is in the region $R_{1}$. The edge $(u_{3},w_{3})$ can only be realized by crossing exactly one edge incident to $u_{1}$, say $(u_{1},w_{5})$, and exactly one edge incident to $u_{2}$, say $(u_{2},w_{4})$ (see dashed red edge in Fig. 21). Now the edge $(u_{3},w_{4})$ is not realizable. Namely, we are not allowed to draw this edge through the regions $R_{2}$, $R_{3}$ and $R_{4}$, since in this case it would cross at least two edges incident to $u_{1}$ or at least two edges incident to $u_{2}$ – a fan-crossing. Also, it cannot cross the edge $(u_{1},w_{5}$), as this would also create a fan-crossing. The only option left is to cross the edge $(u_{2},w_{5})$; however, this implies that $(u_{3},w_{4})$ crosses $(u_{3},w_{3})$, which is not allowed. Hence, $\Gamma_{1}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{2}$. In drawing $\Gamma_{2}$ we have six regions, denoted by $R_{1},\ldots,R_{6}$ as in Fig. 22(a). We observe some consequences of the previous lemmas:

(i) vertices $u_{3},u_{4},u_{5}$ can lie neither in $R_{2},R_{3}$ (Lemma 2) nor in $R_{4}$ (Lemma 3);

(ii) vertices $u_{3},u_{4},u_{5}$ cannot lie in $R_{5}$, since the edge connecting one of them to $w_{1}$ would cross both edges $(u_{1},w_{4})$ and $(u_{1},w_{5})$ (Lemma 3);

(iii) at most one of vertices $u_{3},u_{4},u_{5}$ can lie in $R_{6}$ (Lemma 3).

By the previous analysis, at least two vertices, say $u_{3}$ and $u_{4}$, lie in region $R_{1}$. At most one of the edges $(u_{3},w_{3})$ and $(u_{4},w_{3})$ can cross $(u_{1},w_{2})$ and $(u_{2},w_{1})$. Thus, at least one of these edges has to cross either $(u_{1},w_{4})$ and $(u_{2},w_{5})$, or $(u_{1},w_{5})$ and $(u_{2},w_{4})$, assume w.l.o.g. the latter pair; see Fig. 22(b). However, this implies that it is no longer possible to draw the edge $(u_{3},w_{4})$ fan-crossing free. Hence, $\Gamma_{2}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{3}$. In drawing $\Gamma_{3}$ we have seven regions, denoted by $R_{1},\ldots,R_{7}$ as in Fig. 23.

We observe some consequences of the previous lemmas:

(i) vertices $u_{3},u_{4},u_{5}$ can lie neither in $R_{3}$ nor in $R_{5}$ (Lemma 2);

(ii) vertices $u_{3},u_{4},u_{5}$ cannot lie in $R_{6}$, since the edge connecting one of them to $w_{1}$ would cross both edges $(u_{1},w_{4})$ and $(u_{1},w_{5})$ (Lemma 3);

(iii) vertices $u_{3},u_{4},u_{5}$ cannot lie in $R_{7}$, since the edges connecting one of them to $w_{1}$ and $w_{2}$ would both cross $(u_{1},w_{3})$ (Lemma 3).

Since $u_{3},u_{4},u_{5}$ are all connected to $w_{1}$ and $w_{2}$, at least one of them, say $u_{3}$, must lie in region $R_{2}$. First we consider the edge $(u_{3},w_{3})$. This edge can neither cross $(u_{1},w_{3})$, nor $(u_{2},w_{2})$, nor both $(u_{1},w_{1})$ and $(u_{1},w_{2})$, nor both $(w_{1},u_{1})$ and $(w_{1},u_{2})$. So, the only option for edge $(u_{3},w_{3})$ is to cross $(u_{1},w_{2})$ (see Fig. 23). However, since the edges $(u_{3},w_{4})$ and $(u_{3},w_{5})$ can cross neither $(u_{1},w_{2})$ nor $(u_{3},w_{3})$, they both cross $(u_{1},w_{3})$, making a fan-crossing. Hence, $\Gamma_{3}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{4}$. In drawing $\Gamma_{4}$ we have seven regions, denoted by $R_{1},\ldots,R_{7}$, as in Fig. 24.

By the lemmas above we conclude that vertices $u_{3},u_{4},u_{5}$ can lie neither in $R_{2},R_{3},R_{5},R_{6}$ (Lemma 2) nor in $R_{4}$ (Lemma 3). Also, at most one of them can lie in $R_{1}$ and $R_{7}$ (Lemma 3). So we cannot place all of the three nodes $u_{3},u_{4},u_{5}$ together with their edges and obtain a fan-crossing free drawing. Hence, $\Gamma_{4}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{5}$. In drawing $\Gamma_{5}$ we have eight regions, denoted by $R_{1},\ldots,R_{8}$ as in Fig. 25.

The lemmas above imply that:

(i) vertices $u_{3},u_{4},u_{5}$ can lie neither in $R_{3},R_{5},R_{7},R_{8}$ (Lemma 2) nor in $R_{6}$ (Lemma 2);

(ii) vertices $u_{3},u_{4},u_{5}$ cannot lie in $R_{4}$, since the edges connecting one of them, say $u_{3}$, to $w_{4}$ and $w_{5}$ would cross $(u_{1},w_{2})$ and $(u_{2},w_{2})$, respectively. But then edge $(u_{3},w_{4})$ is not allowed to cross $(u_{1},w_{3})$ and $(u_{2},w_{2})$, and so this edge is not drawable at all;

(iii) vertices $u_{3},u_{4},u_{5}$ cannot lie in $R_{2}$, since the edges connecting one of them, say $u_{3}$, to $w_{4}$ and $w_{5}$ would cross $(u_{1},w_{3})$ and $(u_{2},w_{3})$, respectively. But then edge $(u_{3},w_{5})$ is not allowed to cross $(u_{1},w_{3})$ and $(u_{2},w_{2})$, and so this edge is not drawable at all.

Thus, the only option left is that all the vertices $u_{3},u_{4}$, and $u_{5}$ are in $R_{1}$. However, edges $(u_{3},w_{3})$, $(u_{4},w_{3})$ and $(u_{5},w_{3})$ are neither allowed to cross $(u_{2},w_{2})$, nor to cross both $(u_{1},w_{3})$ and $(u_{1},w_{2})$, nor to cross both $(u_{2},w_{4})$ and $(u_{2},w_{1})$, nor to cross both $(u_{1},w_{4})$ and $(u_{1},w_{5})$, and so they cannot be drawn. Hence, $\Gamma_{5}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{6}$. In drawing $\Gamma_{6}$, we have seven regions, denoted by $R_{1},\ldots,R_{7}$ as in Fig. 26.

The lemmas above imply that:

(i) vertices $u_{3},u_{4},u_{5}$ can lie neither in regions $R_{2},R_{3},R_{5},R_{6}$ (Lemma 2) nor in region $R_{7}$ (Lemma 3);

(ii) vertices $u_{3},u_{4},u_{5}$ cannot lie in $R_{4}$, since the edge connecting one of them to $w_{1}$ (to $w_{2}$) would cross both edges $(u_{1},w_{3})$ and $(u_{1},w_{4})$ (both edges $(u_{2},w_{3})$ and $(u_{2},w_{4})$), hence creating a fan-crossing.

The only option left is that all vertices $u_{3},u_{4}$ and $u_{5}$ are in $R_{1}$. However, the edge connecting one of them to $w_{3}$ can neither cross $(u_{1},w_{4})$, nor $(u_{2},w_{3})$, nor the pair $(u_{1},w_{2})$ and $(u_{1},w_{1})$, nor the pair $(u_{2},w_{1})$ and $(u_{2},w_{2})$. So the only option for this edge is to cross the edges $(u_{1},w_{2})$ and $(u_{2},w_{1})$. However, this cannot be done for all the three vertices. Hence, $\Gamma_{6}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

The drawing $\Gamma_{7}$. In drawing $\Gamma_{7}$, we have seven regions, denoted by $R_{1},\ldots,R_{7}$ as in Fig. 27.

By the lemmas above we get that vertices $u_{3},u_{4},u_{5}$ can lie neither in regions $R_{2},R_{3},R_{6},R_{7}$ (Lemma 2) nor in regions $R_{4},R_{5}$ (Lemma 3). Thus, the only option left is that all of them are in $R_{1}$. However, the edge connecting one of them to $w_{3}$ can cross neither both edges $(u_{1},w_{2})$ and $(u_{1},w_{1})$, nor both edges $(u_{2},w_{1})$ and $(u_{2},w_{2})$, not both edges $(u_{1},w_{5})$ and $(u_{1},w_{4})$, nor both edges $(u_{2},w_{4})$ and $(u_{2},w_{5})$. Hence each of such edges must cross either both edges $(u_{1},w_{2})$ and $(u_{2},w_{1})$, or both edges $(u_{1},w_{5})$ and $(u_{2},w_{4})$. Since there are three such edges, there must be a fan-crossing. Hence, $\Gamma_{7}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{5,5}$.

So we can conclude that there is no fan-crossing free drawing of $K_{5,5}$.

Appendix 0.G Graph $K_{3,7}$ is not fan-crossing free

We now turn our attention to graph $K_{3,7}$. As in Section 0.F, we consider each of the seven drawings $\Gamma_{1},\ldots,\Gamma_{7}$ of $K_{2,5}$ separately and try to add more vertices to them without violating the fan-crossing free property.

The drawing $\Gamma_{1}$. We already proved that in the presence of vertex $u_{3}$, drawing $\Gamma_{1}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,5}$.

The drawing $\Gamma_{2}$. In drawing $\Gamma_{2}$, we have six regions $R_{1},\ldots,R_{6}$ as in Fig. 22(a). As in the corresponding case of Section 0.F, vertex $u_{3}$ can lie neither in $R_{2},R_{3}$ (Lemma 2), nor in $R_{4}$ (Lemma 3), nor in $R_{5}$.

First, we assume that $u_{3}$ is in the region $R_{1}$. Then, similarly to the corresponding case of Section 0.F for $K_{5,5}$, it is not possible to draw $(u_{3},w_{3})$ through the regions $R_{5}$ and $R_{6}$. So, $(u_{3},w_{3})$ must cross $(u_{1},w_{2})$ and $(u_{2},w_{1})$; also, the edge $(u_{3},w_{4})$ must cross one of $(u_{1},w_{5})$ or $(u_{2},w_{5})$, say the former; see Fig. 28(a), in which the regions $R^{\prime}_{1}$ and $R^{\prime}_{6}$ are the faces delimited by the black and red lines in the plane graph.

Now, consider vertex $w_{6}\in V_{2}$. This vertex cannot be in region $R_{2}$. Indeed, if $w_{6}$ were in $R_{2}$, then both edges $(u_{2},w_{6})$ and $(u_{3},w_{6})$ must cross $(u_{1},w_{1})$, since they are allowed to cross neither $(u_{2},w_{1})$ nor $(u_{1},w_{2})$; this yields a fan-crossing. Also, vertex $w_{6}$ cannot be in $R_{3}$, due to a similar observation. If $w_{6}$ were in $R_{3}$, then both edges $(u_{1},w_{6})$ and $(u_{3},w_{6})$ must cross $(u_{2},w_{2})$, since they are allowed to cross neither $(u_{2},w_{1})$ nor $(u_{1},w_{2})$, which again yields a fan-crossing. Finally, $w_{6}$ cannot be in one of the regions $R_{4}$ or $R_{5}$, as otherwise the edge from $u_{3}$ to $w_{6}$ cannot be drawn without introducing a fan-crossing. Thus, $w_{6}$ can be only in either $R_{6}$ or $R_{1}$. We consider each of these two cases separately.

First, consider the case in which $w_{6}$ is in $R_{6}$. Since $(u_{3},w_{6})$ is allowed to cross neither $(u_{1},w_{5})$ nor $(u_{3},w_{4})$, and since this edge cannot cross $(u_{1},w_{4})$ without introducing a fan-crossing, vertex $w_{6}$ must be in $R^{\prime}_{6}$. To avoid introducing any fan-crossing, edge $(u_{3},w_{6})$ must cross $(u_{2},w_{3})$. Similarly, the edge $(u_{1},w_{6})$ must cross $(u_{2},w_{4})$; see dashed dotted blue edges in Fig. 28(b).

Next, we argue for vertex $w_{7}$, which cannot lie in $R^{\prime}_{6}$, since by the arguments above $(u_{3},w_{7})$ would have to cross $(u_{2},w_{5})$. So, $w_{7}$ must be in $R_{1}$. It is not difficult to see that $(u_{1},w_{7})$ cannot cross an edge incident to $u_{1}$, or one of the edges $(u_{3},w_{3})$, $(u_{3},w_{4})$ and $(u_{2},w_{4})$, or the pair $(u_{2},w_{3})$ and $(u_{2},w_{1})$. As a result, $w_{7}$ must be in $R^{\prime}_{1}$. But then it is easy to see that $(u_{2},w_{7})$ yields inevitably a fan-crossing. In fact, this edge is not allowed to cross any edge incident to $u_{2}$, or one of the edges $(u_{3},w_{3})$, $(u_{3},w_{6})$ and $(u_{1},w_{6})$, or both edges $(u_{1},w_{3})$, $(u_{1},w_{2})$.

It remains to consider the case in which $w_{6}$ in $R_{1}$. We claim that in this case $w_{6}$ is inevitably in $R^{\prime}_{1}$. To see this, observe that $(u_{1},w_{6})$ can cross neither any edge incident to $u_{1}$, nor one of the edges $(u_{3},w_{3})$ and $(u_{3},w_{4})$, nor both edges $(u_{2},w_{4})$ and $(u_{2},w_{5})$, nor both edges $(u_{2},w_{3})$ and $(u_{2},w_{1})$. As in the previous case, we next argue for $w_{7}$, which cannot be in $R_{6}$ (by the arguments above). So, we can assume that $w_{7}$ is in $R_{1}$, too. By the same arguments as for $w_{6}$, we obtain that $w_{7}$ is in $R^{\prime}_{1}$. For both edges $(u_{2},w_{6})$ and $(u_{2},w_{7})$ the following holds. They can cross neither $(u_{3},w_{3})$, nor the pair $(u_{1},w_{3})$ and $(u_{1},w_{2})$, nor the pair $(u_{1},w_{4})$ and $(u_{1},w_{5})$. Thus, both have to cross $(u_{3},w_{4})$, yielding a fan-crossing.

From the case analysis above, we conclude that in the case in which $u_{3}$ is in $R_{1}$, it is not possible to augment $\Gamma_{2}$ to a fan-crossing free drawing of $K_{3,7}$.

We now consider the second case of our analysis for $u_{3}$, in which $u_{3}$ is in $R_{6}$. By Lemma 3, the edges $(u_{3},w_{1})$ and $(u_{3},w_{2})$ cross the edges $(u_{1},w_{5})$ and $(u_{2},w_{5})$, respectively; see Fig 29(a). It follows that $(u_{3},w_{3})$ crosses either $(u_{1},w_{4})$ or $(u_{2},w_{4})$. W.l.o.g. assume that $(u_{3},w_{3})$ crosses $(u_{1},w_{4})$. Then, $(u_{3},w_{4})$ must be planar, as otherwise regardless of its drawing a fan-crossing is introduced.

We next argue for $w_{6}$. This vertex can be neither in $R_{2}$ nor in $R_{3}$, as otherwise either the edge $(u_{2},w_{6})$ or the $(u_{1},w_{6})$ would yield a fan-crossing, respectively. Vertex $w_{6}$ can be neither in $R_{1}$ nor in $R_{4}$ as well, because in both cases edge $(u_{3},w_{6})$ yields a fan-crossing. In follows that $w_{6}$ is either in $R_{5}$ or in $R_{6}$. We consider each of these two cases separately in the following. First suppose that $w_{6}$ is in $R_{6}$. The edges incident to $u_{3}$ partition $R_{6}$ into three subregions, which we denote by $R_{6}^{(1)}$, $R_{6}^{(2)}$ and $R_{6}^{(3)}$; refer to the regions annotated with red labels in Fig 29(b). If $w_{6}$ is in $R_{6}^{(2)}$, then both edges $(u_{2},w_{6})$ and $(u_{3},w_{6})$ can be even crossing-free, but nevertheless $(u_{1},w_{6})$ yields a fan-crossing. If $w_{6}$ is in $R_{6}^{(3)}$, then the edges $(u_{1},w_{6})$ and $(u_{2},w_{6})$ yield a fan-crossing. It follows that $w_{6}$ is in $R_{6}^{(1)}$.

We next observe that $(u_{1},w_{6})$ can cross none of the edges $(u_{1},w_{4})$, $(u_{1},w_{5})$, $(u_{3},w_{1})$ and $(u_{3},w_{3})$, which implies that $(u_{1},w_{6})$ is crossing-free; see dashed-dotted blue edged in Fig 29(b)). Analogously, $(u_{3},w_{6})$ must be also crossing-free. Now consider the edge $(u_{2},w_{6})$. Observe that this edge cannot cross an edge incident to $u_{2}$, or two edges incident to $u_{1}$, or two edges incident to $u_{3}$. We conclude that the only way to draw $(u_{2},w_{6})$ is by crossing $(u_{1},w_{4})$ and $(u_{3},w_{3})$.

On the other hand, vertex $w_{7}$ cannot lie in $R_{6}$, since the same arguments as for $w_{6}$ above imply that $(u_{2},w_{7})$ must cross $(u_{1},w_{4})$ and $(u_{3},w_{3})$, thus forming a fan-crossing. So, vertex $w_{7}$ is in $R_{5}$. In particular, in the presence of the edge $(u_{3},w_{7})$, vertex $w_{7}$ must be in the region denoted by $R^{\prime}_{5}$ (see Fig 29(b)). However, this implies that the edge $(u_{1},w_{7})$ yields a fan-crossing; indeed, this edge cannot cross $(u_{3},w_{3})$, or any edge incident to $u_{1}$, or the pair of edges $(u_{2},w_{6})$ and $(u_{2},w_{3})$, or the pair $(u_{2},w_{4})$ and $(u_{2},w_{5})$.

It remains to consider the case in which $w_{6}$ is in $R_{5}$. In the presence of the edge $(u_{3},w_{3})$, this region is partitioned into two subregions $R_{5}^{(1)}$ and $R_{5}^{(2)}$; refer to the regions annotated with red labels in Fig 30. Suppose first that $w_{6}$ is in $R_{5}^{(1)}$. Then, it is not difficult to see that $(u_{3},w_{6})$ inevitably yields a fan-crossing, as this edge cannot cross $(u_{1},w_{4})$ or $(u_{3},w_{3})$.It follows that $w_{6}$ must be in $R_{5}^{(2)}$. In this case, the edge $(u_{1},w_{6})$ has to cross $(u_{2},w_{3})$. Also, the edge $(u_{3},w_{6})$ has to cross $(u_{2},w_{4})$, and the edge $(u_{2},w_{6})$ must be crossing-freee; see dashed-dotted blue edges in Fig 30). Now, observe that $R_{5}$ cannot contain any other vertex among those that are not in the drawing constructed so far, since the edges incident to this vertex would cross exactly the same edges as the edges incident to $w_{6}$, which inevitably yield fan-crossings. Since these vertices cannot also lie in $R_{6}$ (as we consider this case earlier), we conclude that $\Gamma_{2}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$, when $u_{3}$ is in $R_{6}$.

Our analysis for vertex $u_{3}$ suggests that $\Gamma_{2}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$.

The drawing $\Gamma_{3}$. In drawing $\Gamma_{3}$ we have seven regions $R_{1},\ldots,R_{7}$ as in Fig. 23. As proved in Section 0.F, vertex $u_{3}$ can lie neither in regions $R_{3},R_{5}$ (Lemma 2), nor in regions $R_{6},R_{7}$ (Lemma 3). Thus, vertex $u_{3}$ must lie in one of the following regions: $R_{2}$, $R_{4}$ or $R_{1}$. We consider each of these case separately.

We first consider the case in which $u_{3}$ is in $R_{4}$. By Lemma 3, $w_{6}$ must be either in $R_{4}$ or in $R_{5}$. Both cases are not possible, because edge $(u_{1},w_{6})$ inevitably yields a fan-crossing (in fact, this edge cannot cross any of the edges $(u_{1},w_{2})$, $(u_{2},w_{1})$, or $(u_{2},w_{2})$). This rules out the case in which $u_{3}$ is in $R_{4}$.

Next, we consider the case in which $u_{3}$ is in $R_{2}$. By Lemma 3, $w_{6}$ must be either in $R_{2}$ or in $R_{3}$. Both cases are not possible, because edge $(u_{2},w_{6})$ inevitably yields a fan-crossing (in fact, this edge cannot cross any of the edges $(u_{2},w_{2})$, $(u_{1},w_{3})$, or $(u_{1},w_{2})$). So, the case in which $u_{3}$ is in $R_{2}$ is also ruled out.

Finally, we consider the case in which $u_{3}$ is in $R_{1}$. By Lemma 3 (for $R=R_{1}$ [$R=R_{7}$], $w_{i}=w_{2}$, $w_{j}=w_{3}$ and $w_{k}=w_{5}$ [$w_{k}=w_{4}$]), it follows that the edge $(u_{3},w_{2})$ has to cross the pair of edges $(u_{1},w_{5})$ and $(u_{1},w_{4})$, which yields a fan-crossing. As a consequence, $u_{3}$ cannot be in $R_{1}$.

Our case analysis on $u_{3}$ implies that $\Gamma_{3}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$.

The drawing $\Gamma_{4}$. Next, we consider drawing $\Gamma_{4}$, which has seven regions $R_{1},\ldots,R_{7}$ as in Fig. 24. As proved in Section 0.F, vertex $u_{3}$ can lie neither in regions $R_{2},R_{3},R_{5},R_{6}$ (Lemma 2), nor in region $R_{4}$ (Lemma 3). This implies that $u_{3}$ is either in $R_{1}$ or in $R_{7}$. Suppose that $u_{3}$ is in $R_{7}$; the case where $u_{3}$ is in $R_{1}$ is symmetric. By Lemma 3, the edges $(u_{3},w_{1})$ and $(u_{3},w_{2})$ cross $(u_{1},w_{5})$ and $(u_{2},w_{5})$, respectively; see Fig. 31(a). It follows that the edges $(u_{3},w_{3})$, $(u_{3},w_{4})$ and $(u_{3},w_{5})$ must be crossing-free.

Next, we argue for vertex $w_{6}\in V_{2}$. It is immediate to see that this vertex cannot lie in one of the regions $R_{2},R_{3},R_{5}$ and $R_{6}$, as otherwise either the edge $(u_{1},w_{6})$ or the edge $(u_{2},w_{6})$ yields a fan-crossing. We next claim that $w_{6}$ is not in $R_{1}$, as well. Indeed, if $w_{6}$ were in $R_{1}$, then the edge $(u_{3},w_{6})$ would create a fan-crossing. It follows that $w_{6}$ is either in $R_{4}$ or in $R_{7}$.

First, we consider the case in which $w_{6}$ is in $R_{7}$. Observe that the edges incident to $u_{3}$ partition the region $R_{7}$ into $5$ new regions, denoted by $R_{7}^{(1)},\ldots R_{7}^{(5)}$; see Fig. 31(b). If $w_{6}$ is in $R_{7}^{(2)}$, then it is easy to both two edges $(u_{1},w_{6})$ and $(u_{3},w_{6})$ can be present, this is not true for the edge $(u_{2},w_{6})$, which inevitably yields a fan-crossing; in fact, the edge $(u_{2},w_{6})$ cannot cross an edge incident to $u_{2}$, or $(u_{3},w_{2})$, or $(u_{1},w_{4})$, or two edges incident to $u_{1}$. This already implies that $(u_{2},w_{6})$ must cross $(u_{3},w_{4})$ and $(u_{3},w_{3})$, which is not allowed. So, $w_{6}$ cannot be in $R_{7}^{(2)}$. By symmetry, $w_{6}$ cannot be in $R_{7}^{(5)}$ either. If $w_{6}$ is in $R_{7}^{(3)}$, then the edge $(u_{1},w_{6})$ inevitably yields a fan-crossing; indeed, $(u_{1},w_{6})$ cannot cross an edge incident to $u_{1}$ (especially the edge $(u_{1},w_{5})$), or the edge $(u_{3},w_{1})$, nor two edges incident to $w_{5}$ (namely, the edges $(u_{3},w_{5})$ and $(u_{2},w_{5})$), or two edges incident to $u_{3}$ (namely, the edges $(u_{3},w_{5})$ and $(u_{2},w_{2})$. Hence, $w_{6}$ can be neither in $R_{7}^{(3)}$ nor in $R_{7}^{(4)}$ (by symmetry). Fig. 32(a) illustrates the case in which $w_{6}$ is in $R_{7}^{(1)}$. In this case, we first observe that $(u_{1},w_{6})$ must cross $(u_{3},w_{3})$. Also, $(u_{2},w_{6})$ must cross $(u_{3},w_{4})$. Hence, $(u_{3},w_{6})$ must be crossing-free. Since it is easy to see that $w_{7}$ can be neither in $R_{7}$ nor in $R_{4}$, the case in which $w_{6}$ is in $R_{7}$ is ruled out.

To complete the analysis of drawing $\Gamma_{4}$, it remains to consider the case, in which $w_{6}$ is in $R_{4}$. In this case, the edge $(u_{3},w_{6})$ has to cross both edges $(u_{1},w_{4})$ and $(u_{2},w_{3})$. In addition, both edges $(u_{1},w_{6})$ and $(u_{2},w_{6})$ must be crossing-free; see dashed-dotted blue edges in Fig. 32(b). We next argue for $w_{7}$, which by symmetry can only be in $R_{4}$. However, in this case, $(u_{3},w_{7})$ would yield a fan-crossing. Since $w_{6}$ can be neither in $R_{7}$ nor in $R_{4}$, we conclude that the drawing $\Gamma_{4}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$.

The drawing $\Gamma_{5}$. In drawing $\Gamma_{5}$, we have eight regions $R_{1},\ldots,R_{8}$ as in Fig. 25. As proved in Section 0.F, vertex $u_{3}$ can lie neither in regions $R_{3},R_{5},R_{7},R_{8}$ (Lemma 2), nor in regions $R_{1},R_{6}$ (Lemma 3). It follows that vertex $u_{3}$ lies either in $R_{2}$ or in $R_{4}$.

Suppose first that $u_{3}$ lies in $R_{4}$. We next argue for vertex $w_{6}\in V_{2}$. By Lemma 3, vertex $w_{6}$ must be in $R_{4}$ or $R_{5}$. Both cases, however, are not possible, as the edge $(u_{1},w_{6})$ inevitably yields a fan-crossing; in fact, this edge cannot cross any of the edges $(u_{1},w_{2})$, $(u_{2},w_{1})$, and $(u_{2},w_{2})$, and as a results when $u_{3}$ is in $R_{4}$, a fan-crossing is always yielded.

Suppose now that $u_{3}$ is in $R_{2}$. Again we continue by arguing for $w_{6}$. By Lemma 3, this vertex must be either in $R_{2}$ or $R_{3}$. However, again both cases are not possible, as the edge $(u_{2},w_{6})$ inevitably yields a fan-crossing; in fact, this edge is cannot cross any of the edges $(u_{2},w_{2})$, $(u_{1},w_{3})$, and $(u_{1},w_{2})$, and as a result when $u_{3}$ is in $R_{2}$, a fan-crossing is always yielded. We conclude that the drawing $\Gamma_{5}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$.

The drawing $\Gamma_{6}$. In drawing $\Gamma_{6}$, we have seven regions $R_{1},\ldots,R_{7}$ as in Fig. 26. As proved in Section 0.F, vertex $u_{3}$ can lie neither in regions $R_{2},R_{3},R_{5},R_{6}$ (Lemma 2), nor in regions $R_{1},R_{4},R_{7}$ (Lemma 3). Hence, in this case we can directly conclude that the drawing $\Gamma_{5}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$.

The drawing $\Gamma_{7}$. In drawing $\Gamma_{7}$, we have seven regions $R_{1},\ldots,R_{7}$ as in Fig. 27. As proved in Section 0.F, vertex $u_{3}$ can lie neither in regions $R_{2},R_{3},R_{6},R_{7}$ (Lemma 2), nor in regions $R_{4},R_{5}$ (Lemma 3).

Hence, vertex $u_{3}$ can only be in $R_{1}$. The edge $(u_{3},w_{3})$ can be drawn without introducing fan-crossings only if it crosses either both edges $(u_{1},w_{2})$ and $(u_{2},w_{1})$, or both edges $(u_{1},w_{5})$ and $(u_{1},w_{4})$. Assume w.l.o.g. the former. Then, each of the edges $(u_{3},w_{1})$, $(u_{3},w_{2})$, $(u_{3},w_{4})$ and $(u_{3},w_{5})$ must be crossing-free; see Fig. 33(a). We next argue for $w_{6}\in V_{2}$. We first observe that $w_{6}$ cannot lie in one of the regions $R_{2},R_{3},R_{6}$ or $R_{7}$, since then the edge $(u_{2},w_{6})$ (if $w_{6}$ is in $R_{2}$ or $R_{3}$) or the edge $(u_{1},w_{6})$ (if $w_{6}$ is in $R_{6}$ or $R_{7}$) would yield fan-crossings. Also, $w_{6}$ is not in $R_{4}$, as otherwise $(u_{3},w_{6})$ would yield fan-crossings. Thus, $w_{6}$ must be either in $R_{1}$ or in $R_{5}$ (by symmetry, $w_{7}$ must be either in $R_{1}$ or in $R_{5}$).

If $w_{6}\in V_{2}$ is in $R_{1}$, then it must be placed in the outer face of the drawing containing $u_{3}$, as otherwise one of the edges $(u_{1},w_{6})$ or $(u_{2},w_{6})$ would yield a fan-crossing. Further, the edge $(u_{1},w_{6})$ must cross $(u_{3},w_{4})$ and the edge $(u_{2},w_{6})$ must cross $(u_{3},w_{5})$. This implies that there $w_{6}$ is the only vertex in $R_{1}$, and the edge $(u_{3},w_{6})$ must be crossing-free; see Fig. 33(b). As a result, $w_{7}$ is in $R_{5}$ (recall that $w_{7}$ can either be in $R_{1}$ or in $R_{5}$). In this case, $(u_{3},w_{7})$ has to cross both edges $(u_{1},w_{5})$ and $(u_{2},w_{4})$, and additionally one of the edges $(u_{1},w_{6})$ or $(u_{2},w_{6})$, which yields a a fan-crossing. This rules out the case, in which $w_{6}\in V_{2}$ is in $R_{1}$.

To complete the case analysis, it remains to consider the case in which $w_{6}$ is in $R_{5}$. In this case, $(u_{3},v)$ must cross both edges $(u_{1},w_{5})$ and $(u_{2},w_{4})$. This already implies that there $w_{6}$ is the only vertex in $R_{5}$ and that each of the edges $(u_{1},w_{6})$ and $(u_{2},w_{6})$ must be drawn crossing-free; see Fig. 34. However, in this case $w_{7}$ must be in $R_{1}$, which yields a case that we have already ruled out. We conclude that the drawing $\Gamma_{7}$ cannot be a subdrawing of a fan-crossing free drawing of $K_{3,7}$, which concludes the proof of Characterization 4.8.