Visualizing Treewidth

TL;DR

This paper presents visualization techniques for graph witness drawings that effectively display bounded treewidth or pathwidth, using tree decompositions, optimized layouts, and various drawing styles to minimize crossings.

Contribution

It introduces new visualization paradigms for witness drawings of graphs with bounded treewidth, including a taxonomy of styles and algorithms for crossing minimization.

Findings

Implemented a prototype for crossing minimization using dynamic programming and heuristics.

Developed a taxonomy of drawing styles for witness graphs, including arc, circular, and straight-line layouts.

Demonstrated effective visualization of graphs with small and large treewidth.

Abstract

A witness drawing of a graph is a visualization that clearly shows a given property of a graph. We study and implement various drawing paradigms for witness drawings to clearly show that graphs have bounded pathwidth or treewidth. Our approach draws the tree decomposition or path decomposition as a tree of bags, with induced subgraphs shown in each bag, and with ''tracks'' for each graph vertex connecting its copies in multiple bags. Within bags, we optimize the vertex layout to avoid crossings of edges and tracks. We implement a visualization prototype for crossing minimization using dynamic programming for graphs of small width and heuristic approaches for graphs of larger width. We introduce a taxonomy of drawing styles, which render the subgraph for each bag as an arc diagram with one or two pages or as a circular layout with straight-line edges, and we render tracks either with straight lines or with orbital-radial paths.

Figures

Equations

$$C[S_i]=\{\begin{array}{ll}\text{CrossingCountEE}{S}_i& \text{if}i=1\\ \underset{\begin{array}{c}{S}_{i-1}=(i-1,{\text{Drawing}}_{i-1},\Delta ({\text{Drawing}}_{i-1})),\\ {\text{Drawing}}_{i-1}\in {\mathcal{G}}_{i-1},\Delta ({\text{Drawing}}_{i-1})\in \mathcal{D}({\text{Drawing}}_{i-1})\end{array}}{\min }C[S_{i-1}]+\text{CrossingCount}{S}_i,S_{i-1}& \text{otherwise}\end{array}$$

$$\text{CrossingCountEE}{S}_i$$

$$\text{CrossingCountTE}{S}_i,S_{i-1}$$

$$+\sum _{\begin{array}{c}(u,v)\in E_i\\ {\sigma }_i((u,v))=\ell \end{array}}\left|\{w\in V_{i-1}\cap V_i\mid u{\prec }_{i}w{\prec }_{i}v\}\right|$$

$$\text{CrossingCountTT}{S}_i,S_{i-1}$$

$$C[S_i]=\{\begin{array}{ll}\text{CrossingCountEE}{S}_i& \text{if}{\mathrm{deg}}^{-}(V_i)=0\\ \underset{\begin{array}{c}{S}_x=(x,{\text{Drawing}}_x,\Delta ({\text{Drawing}}_x))\\ V_x\in N^{-}(V_i)\end{array}}{\min }C[S_x]+\text{CrossingCount}{S}_i,S_x& \text{if}{\mathrm{deg}}^{-}(V_i)=1\\ \underset{\begin{array}{c}(S_x,S_y)\\ V_x,V_y\in N^{-}(V_i),x\ne y\end{array}}{\min }C[S_x]+C[S_y]+\text{CrossingCount}{S}_i,S_x,S_y& \text{if}{\mathrm{deg}}^{-}(V_i)=2\end{array}$$

$$\text{CrossingCountTE}{S}_i,S_x,S_y$$

$$\displaystyle+\sum_{\begin{subarray}{c}(u,v)\in E_{i}\\ \sigma_{i}((u,v))=r\end{subarray}}\left|\{w\in V_{i^{\prime}}\cap V_{i}\mid u\prec_{i}w\prec_{i}v\}\right|\Bigg{)}$$

$$\text{CrossingCountTT}{S}_i,S_x,S_y$$

$$+\sum _{i^{\prime }\in \{x,y\}}\left|\{u,v\in V_{i^{\prime }}\cap V_i\mid u{\prec }_{i^{\prime }}v,v{\prec }_{i}u\}\right|$$

Full text

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University of California, Irvine, USAchiua13@uci.eduhttps://orcid.org/0009-0009-6863-859X TU Wien, Vienna, Austriatdepian@ac.tuwien.ac.athttps://orcid.org/0009-0003-7498-6271Supported by Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]. University of California, Irvine, USAeppstein@uci.eduSupported by NSF grant 2212129. University of California, Irvine, USAgoodrich@uci.eduhttps://orcid.org/0000-0002-8943-191XSupported by NSF grant 2212129. TU Wien, Vienna, Austrianoellenburg@ac.tuwien.ac.athttps://orcid.org/0000-0003-0454-3937Supported by Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]. \CopyrightAlvin Chiu, Thomas Depian, David Eppstein, Michael T. Goodrich, Martin Nöllenburg \ccsdesc[500]Theory of computation Design and analysis of algorithms \ccsdesc[500]Theory of computation Dynamic programming \ccsdesc[500]Human-centered computing Graph drawings \supplementdetailsSource Codehttps://doi.org/10.17605/OSF.IO/QFZ5V \EventEditorsVida Dujmović and Fabrizio Montecchiani \EventNoEds2 \EventLongTitle33rd International Symposium on Graph Drawing and Network Visualization (GD 2025) \EventShortTitleGD 2025 \EventAcronymGD \EventYear2025 \EventDateSeptember 24–26, 2025 \EventLocationNorrköping, Sweden \EventLogo \SeriesVolume357 \ArticleNo15 \onlyShort \NewDocumentCommand\DecWidths o\IfBooleanTF{#1}{\overline{\omega}}{\omega}\IfNoValueF{#2}{{}_{#2}} \NewDocumentCommand\Decompositionm\mathcal{#1} \NewDocumentCommand\Drawingo\Gamma\IfNoValueF{#1}{{}_{#1}} \NewDocumentCommand\SymbolLinearo\textsf{L}\IfNoValueF{#1}{{\textsf{#1}}} \NewDocumentCommand\SymbolCircularC \NewDocumentCommand\SymbolOrbitalO \NewDocumentCommand\SymbolTTCrossing$\textsf{t}/\textsf{t}$ \NewDocumentCommand\SymbolTECrossing$\textsf{t}/\textsf{e}$ \NewDocumentCommand\SymbolEECrossing$\textsf{e}/\textsf{e}$ \NewDocumentCommand\CrossingCountTTm\text{cr}_{\textsf{t}/\textsf{t}}(#1) \NewDocumentCommand\CrossingCountTEm\text{cr}_{\textsf{t}/\textsf{e}}(#1) \NewDocumentCommand\CrossingCountEEm\text{cr}_{\textsf{e}/\textsf{e}}(#1) \NewDocumentCommand\CrossingCountm\text{cr}(#1)

Visualizing Treewidth

Alvin Chiu

  

Thomas Depian

  

David Eppstein

  

Michael T. Goodrich

  

Martin Nöllenburg

1 Introduction

Work in graph drawing can be often viewed as studying paradigms for visualizing a graph, $G$, to clearly illustrate one or more properties of $G$ while also optimizing one or more quality criteria, such as area or number of edge crossings; see, e.g., [18, 54, 25, 35, 20]. This viewpoint is related to the concept of a visual proof, which is a proof given as a visual representation called a witness (or visual certificate); see, e.g., [48, 46, 31, 47]. Ideally, a visual proof should be clear and concise: the property being proven should be immediately discernible simply by examining the witness. A prominent example is a crossing-free drawing as a witness for planarity. In this paper, we are interested in visualizing treewidth.

A tree decomposition of a graph, $G=(V,E)$, is a tree, $T$, with nodes, $V_{1},V_{2},\ldots,V_{k}$, which are called bags, where

Each $V_{i}$ is a subset of $V$ and $\bigcup_{i=1}^{k}V_{i}=V$. 2. 2.

If $v\in V_{i}$ and $v\in V_{j}$, then $v$ is in each bag in the (unique) path in $T$ from $V_{i}$ to $V_{j}$. 3. 3.

For each edge $(v,w)\in E$ there is a bag that contains both $v$ and $w$.

The width of a tree decomposition is one less than the size of its largest bag, and the treewidth of a graph is the smallest width of a tree decomposition. The pathwidth of a graph is the width of a smallest-width tree decomposition whose tree is a path. See, e.g., [13, 43, 37, 26, 22, 23] for more details regarding these topics, including applications in graph drawing and results that many \NP-hard optimization problems can be solved in polynomial time for graphs with bounded treewidth or pathwidth.

We are interested in paradigms for witness drawings of graphs to certify their bounded pathwidth or treewidth. Our approaches are inspired by an example visual proof from Wikipedia that a graph has treewidth 3; see Figure˜1. In this illustration, each bag is drawn as a disk with its member vertices drawn as points inside the disk. In addition, for each vertex, $v\in V$, in this drawing, there is a set of curves, each of which we call a track, that connect the different copies of $v$ in the bags of the tree decomposition, i.e., the support of $v$ in $T$. The set of tracks in $T$ for a vertex $v\in V$ in such a drawing will form a subtree in $T$, which will always be a path if $T$ is itself a path. Note, however, that even if the support for every vertex is a path, this does not imply that $T$ is a path, e.g., as shown in Figure˜1.

To provide a witness drawing for the pathwidth or treewidth of a graph, $G=(V,E)$, we add one more requirement to our visualization, which is missing from the visualization shown in Figure˜1, namely information faithfulness, i.e., that the drawing represents the (entire) graph $G$ and its decomposition $T$. In particular, we require that for each edge, $(v,w)\in E$, we must draw $(v,w)$ as a curve inside each bag $V_{i}$ such that $v,w\in V_{i}$ in the given tree decomposition, $T$, of $G$. Still, we are interested in such witness drawings that minimize edge and track crossings, as in Figure˜1.

To see why this requirement is important, consider a graph $G$ with pathwidth at most \DecWidth. Its interval-width is bounded by $\DecWidth+1$, i.e., $G$ is a subgraph of an interval graph $G^{\prime}$ whose largest clique has size at most $\DecWidth+1$ [17]. Since interval graphs can be represented as closed intervals on the real line, such a representation of $G^{\prime}$ can be seen as a certificate that $G$ has bounded pathwidth. However, we argue that this is not an information faithful witness drawing as it shows $G^{\prime}$ rather than $G$ and, in particular, does not allow a reconstruction of $G$ from the (interval) drawing of $G^{\prime}$.

Related Work.

Mehlhorn et al. [48] and McConnell et al. [46] introduce the paradigm of certifying algorithms, which output their result as well as a concise proof, called a “witness”, that shows that the algorithm produced this output correctly. Subsequent to this work, there has been considerable work on certifying algorithms, including their inclusion in the LEDA and CGAL systems [34]. For example, additional work for witness drawings in graph drawing include work on visualizing proximity properties, such as for Delaunay graphs, Gabriel graphs, and rectangle graphs; see, e.g., [5, 41, 4, 2, 40, 3, 39, 42].

In GD 2024, the GraphTrials framework was introduced, which involves a visual proof for a graph property that can be used in an interactive proof modeled after a bench trial before a judge where a prover establishes that a graph has a given property based on its visualization [31]. Although the authors did not include pathwidth or treewidth in their GraphTrials framework, we feel that our approach to producing witness drawings for graphs with bounded pathwidth or treewidth nevertheless fits into their GraphTrials framework. In particular, in addition to faithfully representing the provided information, we argue that our created witness drawings contain all the necessary information to efficiently certify that the graph at hand has bounded pathwidth or treewidth. Moreover, such drawings can facilitate the formation of a mental model for potential judges. Thus, we believe that the drawings satisfy the required properties for visual certificates as defined in the framework [31].

We are interested in the natural optimization criterion of minimizing edge crossings in a witness drawing for a graph with bounded pathwidth or treewidth, including track-track crossings, edge-edge crossings, and track-edge crossings. In particular, few track-track crossings can aid in verifying that our decomposition satisfies Property 2, i.e., that the bags containing the vertex $v$ induce a connected subgraph of $T$, for all $v\in V$. Furthermore, minimizing edge-edge and track-edge crossings improves the visual quality of the witness drawing and can thus support the verification of the (remaining) properties. Although we are not aware of any prior work on this criterion, minimizing track-track crossings is related to minimizing crossings in storyline drawings [21, 38, 33, 55, 56] and metro maps [10, 1, 6, 30, 50, 11, 52, 51]. Minimization of edge-edge crossings within the bags is related to crossing minimization in linear and circular graph layouts [9, 57, 36, 32, 44, 7, 24].

Multiple past works include individual visualizations of tree decompositions. These include drawings of a tree with each bag shown as a set of vertices within each tree node, separate from any drawing of the graph [13, 43], and drawings of a graph overlaid by bags drawn as regions that surround or connect the vertices [29]. Separately from their applications in treewidth and structural graph theory, tree decompositions with bags of non-constant size have also been used in other ways: for instance, SPQR decompositions are, essentially, tree decompositions of adhesion 2, whose bags induce 3-connected subgraphs, and these have often been visualized as a tree of 3-connected components [19, 49]. However, we are unaware of past works studying visualizations of tree decompositions in any systematic way.

Our Results.

In this paper, we systematize and implement witness drawings for graphs with bounded treewidth or pathwidth. Given a graph, $G=(V,E)$, our approach is to draw the vertices in each bag of a tree decomposition, $T$, of $G$ as points in a disk, either in a circular layout or a linear layout. For each vertex, $v\in V$, we connect each pair of copies of $v$ in adjacent bags in $T$ with a curve we call a track, ideally with all the tracks color coded with the same color (the color for $v$). For each bag, $V_{i}$, in $T$, we draw the edges of the subgraph of $G$ induced by $V_{i}$. In the case of a linear layout of vertices, we draw this subgraph as an arc diagram [57] (either as one-page or two-page book drawing) and in the case of a circular layout of vertices, we draw this subgraph as a circular layout with straight-line edges [9]. Our optimization goal is to minimize the total number of edge crossings, including track-track crossings, edge-edge crossings, and track-edge crossings. See Figure 2 for three different witness drawing styles. We design and implement fixed-parameter-tractable linear-time crossing minimization algorithms for graphs with bounded pathwidth or treewidth.

\onlyShort

Missing details for statements marked by ★ can be found in the full version [ARXIV].

2 Preliminaries

For an integer $p\geq 1$, we use $[p]$ as a shorthand for the set $\{1,2,\ldots,p\}$. Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. All considered graphs are simple and undirected. Throughout this paper, we use $n\coloneqq\left|V\right|$ and $m\coloneqq\left|E\right|$ to refer to the number of vertices and edges of $G$, respectively. For a tree (or path) decomposition $T$ of $G$ with bags $V_{1},V_{2},\ldots,V_{k}$, we let $\DecWidth[T]$, or simply $\DecWidth$ if $T$ is clear from the context, denote its width. Recall $\DecWidth[T]\coloneqq\max_{i\in[k]}\left|V_{i}\right|-1$. We use $\DecWidth*\coloneqq\DecWidth+1$ as a shorthand for the maximum number of vertices in a bag. Without loss of generality, we root the tree $T$ at an arbitrary bag. We use decomposition as a synonym for path and tree decompositions. Furthermore, we assume that the decomposition $T$ consists of $k=\mathcal{O}(n)$ bags and that each bag of $T$ has degree at most three. It is well-known that every graph of treewidth \DecWidthadmits a certifying tree decomposition with $k=\mathcal{O}(n)$ fulfilling this property [45, 14]. In particular, in nice tree decompositions, which have found attention in the design of parameterized (dynamic programming) algorithms [17], every bag has degree at most three. With slight abuse of notation, we write $V_{i}\in T$ to indicate that $V_{i}$ is a bag of $T$. Furthermore, for a bag $V_{i}\in T$, we let $G_{i}\coloneqq G[V_{i}]$ and $E_{i}\coloneqq E(G_{i})$ denote the subgraph of $G$ induced by $V_{i}$ and its edge set, respectively.

3 A Taxonomy of Width-Witness Drawing Styles

Before we describe how to compute a witness drawing for a decomposition $T$ of $G$, we examine the different witness drawing styles illustrated in Figure˜2. In all of them, each bag $V_{i}\in T$ is represented as a disk $D_{i}$ of uniform radius. We consider three different drawing styles that differ in how they arrange vertices inside each bag and how they draw the tracks of the support for each vertex. We use parenthesized letters to refer to each style.

Linear Drawings (\SymbolLinear).

In a linear drawing, as in Figure˜2b, we arrange the vertices along a vertical line inside a disk and draw the edges as arcs to the left or right of the vertices. This drawing style is inspired by one- and two-page book drawings [57, 36]. The tracks for each support are straight-line edges connecting the respective vertices. We use \SymbolLinear[1] and \SymbolLinear[2] to differentiate between one- and two-page book drawings where needed. In an \SymbolLinear[1]-drawing, all arcs, must be either to the right or to the left of the vertices, but this can change between different disks. In an \SymbolLinear[2]-drawing, the arcs can be on both sides of the vertices.

Circular Drawings (\SymbolCircular).

Figure˜2c shows a circular drawing of $T$. There, we arrange the vertices $V_{i}$ along a circle inside the disk $D_{i}$ and draw the edges $E_{i}$ as straight-line chords of said circle. Combinatorially speaking, this is equivalent to a one-page book drawing of $G_{i}$ [12]. As for linear drawings, we draw tracks as straight-line edges.

Orbital Drawings (\SymbolOrbital).

Our last drawing style, of orbital drawings, is inspired by the edge routing in previous drawings with circular vertex placements [8, 15]. It is illustrated in Figure˜2d and uses again a circular arrangement of the vertices. However, we now subdivide the disk $D_{i}$ along the vertices $V_{i}$ into an inner and outer part: In the inside, we draw the edges $E_{i}$ as straight-line chords of the respective circle. The outside, however, the so-called track-routing area, is reserved for tracks. These are aligned as orbits and are not allowed to leave the track-routing area. We allow them to be routed clockwise or counterclockwise around $D_{i}$. This drawing style has no track-edge crossings and all track-track crossings occur inside the track-routing area. While this can result in a cleaner visualization, forcing tracks to stay inside the track-routing area may clutter drawings of decompositions of larger width.

In each drawing style, we evenly distribute the vertices within each disk. For linear drawings the vertex permutation uniquely defines each vertex’s position inside the disk, but for the other two drawing styles this still leaves a free rotation angle. To that end, we let $0\leq\alpha<2\pi/\DecWidth*$ denote the starting angle of the first vertex inside each disk, where $\alpha=0$ corresponds to a placement at twelve o’clock (Figure˜3a). We choose a single $\alpha$ across all disks and we select it so that no interior of a track crosses a vertex independent of the order of the vertices inside each disk. For each bag of a tree decomposition, we evenly distribute its children, if there are any, to the right of its disk (Figure˜3b).

A witness drawing $\Drawing$ of a given decomposition $T$ of $G$ consists of a drawing of (i) the tree (or path) $T$, (ii) each of its bags in one of the above-described drawing styles, and (iii) the support for each vertex of $G$. Depending on the drawing style, we call \Drawingan $\SymbolLinear$-, \SymbolCircular-, or \SymbolOrbital-drawing of $T$, or simply drawing, if the style is clear from the context. The drawing \Drawingmight contain crossings, and we differentiate between the following three crossing types: If two tracks cross, we call it a track-track crossing (or simply \SymbolTTCrossing-crossing), if a track and an edge in a disk cross, it is a track-edge (\SymbolTECrossing) crossing, and if two edges of $G$ cross inside a disk, we call it an edge-edge (\SymbolEECrossing) crossing; recall Figure˜2. If neither of these crossings exist, we call $\Drawing$ planar. Note that we do not count crossings with edges of $T$. We treat \SymbolTTCrossing-, \SymbolTECrossing-, and \SymbolEECrossing-crossings equally throughout the remainder of the paper. However, our algorithms can be readily adapted to weight different types of crossings differently.

4 Linear Drawings

We start our study by considering linear drawings, which visualize each $G_{i}$ as a book drawing with one or two pages and draw the tracks as straight lines. Observe that the drawing of $G_{i}$ is uniquely defined by the linear order $\prec_{i}$ of $V_{i}$ and the function $\sigma_{i}\colon E_{i}\to\{\ell,r\}$, also called page assignment, that assigns each edge to the left or right page, respectively. In this section, we present two algorithms that compute, for a given decomposition $T$ of constant width $\DecWidth$, a crossing-minimal \SymbolLinear-drawing in linear time. \shortLongMinimizing the crossings of an \SymbolLinear-drawing is tightly interwoven with minimizing the \SymbolEECrossing-crossings in the book drawing of each $G_{i}$, $i\in[k]$. The known hardness of the latter problem [44] allows us to show also hardness for the former problem, which justifies our focus on constant-width decompositions.However, we first justify with the following theorem our focus on constant-width decompositions.

{statelater}

figureHardness

Theorem 4.1*.*

Deciding if a path decomposition $T$ has an \SymbolLinear[1]-drawing with at most $c$ crossings is \NP-hard. The same holds true for \SymbolLinear[2]-, \SymbolCircular-, and \SymbolOrbital-drawings.

{prooflater}

ptheoremHardness First, we show the statement for \SymbolLinear[1]-drawings and later argue that it also holds for the other drawing styles. We reduce from the \NP-complete problem One-Page Crossing Minimization, which asks for a given graph $G$ whether it admits a one-page book drawing with at most $c$ crossings [44].

Let $(G,c)$ be an instance of One-Page Crossing Minimization with $G=(V,E)$ and $n=\left|V\right|$. We construct the graph $G^{\prime}=(V^{\prime},E^{\prime})$ by taking the disjoint union of the two graphs $G$ and $K_{n}=(V^{n},E^{n})$, i.e., a complete graph on $n$ vertices. Observe $\left|V^{\prime}\right|=2n$. Let $x$ be an arbitrary vertex of $K_{n}$ and $y$ an arbitrary vertex of $G$; we add to $G^{\prime}$ the edge $(x,y)$. As $G^{\prime}$ contains a clique on $n$ vertices, its treewidth (and thus pathwidth) is at least $n-1$ [17]. The path decomposition $T$ with the bags $V_{1}=V^{n}$, $V_{2}=\{x,y\}$, and $V_{3}=V$, connected in this order, shows that the treewidth (and pathwidth) is at most $n-1$. In particular, $T$ is an optimal decomposition of $G^{\prime}$ with respect to its width. It is known that a one-page book drawing of $K_{n}$ has $\binom{n}{4}$ crossings [53], and, due to symmetry, any linear order of $K_{n}$’s vertices induces this many crossings. We now use this fact to show that $(G,c)$ has a one-page book drawing with at most $c$ crossings if and only if $T$ has an \SymbolLinear[1]-drawing with at most $\binom{n}{4}+c$ crossings.

For the forward direction, let $\prec_{G}$ be a linear order of $V$ that yields a one-page drawing of $G$ with at most $c$ crossings. We construct a drawing \Drawingof $T$ by using an arbitrary order $\prec_{1}$ on $V_{1}$ that starts with $x$, i.e., $x\prec_{1}v$ for every $v\in V^{n}$, $v\neq x$, and set $x\prec_{2}y$ and $\prec_{3}\ =\ \prec_{G}$; see also Figure˜4. Observe that this drawing has no \SymbolTTCrossing- and \SymbolTECrossing-crossings. Thus, there are only \SymbolEECrossing-crossings: $\binom{n}{4}$ in the first, and $c$ in the third bag. For the backward direction, assume that $T$ admits an \SymbolLinear[1]-drawing \Drawingwith at most $\binom{n}{4}+c$ crossings. Without loss of generality, we can assume that the bags $V_{1}$, $V_{2}$, and $V_{3}$ are drawn in this order from left to right. The drawing of $K_{n}$ in the first bag has $\binom{n}{4}$-many \SymbolEECrossing-crossings. Thus, any drawing of $G$ in the third bag can have at most $c$ \SymbolEECrossing-crossings. We can use its linear order $\prec_{3}$ as a witness for $(G,c)$.

The above arguments establish \NP-hardness for \SymbolLinear[1]-drawings. One-page book drawings are equivalent to straight-line drawings where the vertices are placed in convex position [27, 12]. Furthermore, between each pair of adjacent bags there is only one track edge. Therefore, the same construction and reasoning yields also \NP-hardness for \SymbolCircular- and \SymbolOrbital-drawings independent of the value for $\alpha$. For \SymbolLinear[2]-drawings, we can use the known \NP-hardness for determining whether a graph has a two-page book embedding, i.e., a crossing-free two-page book drawing. Using the facts that the two-page crossing number of $K_{n}$ is known [53] and that book drawings are agnostic to cyclic rotations of the linear order $\prec_{G}$, we can use the same construction to obtain hardness for \SymbolLinear[2]-drawings. Since a path decomposition is a restricted form of a tree decomposition, Theorem˜4.1 also extends to the latter type of decompositions.

4.1 Linear Drawings for Path Decompositions

We now focus on the more restrictive setting, where we want to compute a witness drawing of a path decomposition $T$ of $G$ of width $\DecWidth$. Throughout this section, we let the bags of $T$ be ordered as they appear on the path $T$ and assume $V_{1}$ to be the leftmost bag.

All of our algorithms, including those for \SymbolCircular- and \SymbolOrbital-drawings, are based on the insight that, for each bag $V_{i}\in T$, the number of crossings involving edges and/or tracks incident to $V_{i}$ is determined by the drawings of $G_{i}$ in the disk $D_{i}$ and the drawings of the induced subgraphs of neighboring bags. This enables us to employ a dynamic programming (DP) algorithm which processes the path decomposition from left to right and computes a crossing-minimal drawing for the first $i$ bags, for every $i\in[k]$. Since the framework of our algorithms is identical across all drawing styles (and decomposition types), we first describe its general structure and later discuss how it can be adapted to specific drawing styles and tree decompositions.

A General DP Algorithm.

We let $k$ denote the number of bags in $T$ and recall $\DecWidth*=\DecWidth+1$. Let $\Drawing_{i}$ be a drawing of the graph $G_{i}$ in the disk $D_{i}$ for each $i\in[k]$. Let $\mathcal{G}_{i}$ denote the set of all possible drawings $\Gamma_{i}$ of $G_{i}$. All drawing styles admit a combinatorial representation of $\Drawing_{i}$, which implies that $\mathcal{G}_{i}$ is finite for each $i\in[k]$. Let $\Delta(\Drawing_{i})$ denote a drawing of the tracks inside the disk $D_{i}$, given a drawing $\Gamma_{i}$ of $G_{i}$, and let $\mathcal{D}(\Gamma_{i})$ be the set of all such drawings. \SymbolLinear- and \SymbolCircular-drawings represent the tracks as straight lines. Therefore, $\Delta(\Drawing_{i})$ is only relevant for \SymbolOrbital-drawings and we leave it undefined for the other two drawing styles. For this drawing style, we can again represent $\Delta(\Drawing_{i})$ combinatorially, ensuring that $\mathcal{D}(\Gamma_{i})$ is finite. We define $\mathcal{G}_{\max}\coloneqq\max_{i\in[k]}\left|\mathcal{G}_{i}\right|$ and $\mathcal{D}_{\max}\coloneqq\max_{\Drawing_{i}\in\mathcal{G}_{i},i\in[k]}\left|\mathcal{D}(\Gamma_{i})\right|$ as the maximum number of drawings of $G_{1},\ldots,G_{k}$ and their corresponding track drawings, respectively.

In our algorithm, we maintain an $\mathcal{O}(k\cdot\mathcal{G}_{\max}\cdot\mathcal{D}_{\max})$-size table $C$, where $C[i,\Drawing_{i},\Delta(\Drawing_{i})]$ stores the minimum number of crossings of a drawing for the bags $V_{1},V_{2},\ldots,V_{i}$, where $\Drawing_{i}$ is the drawing of $G_{i}$ in $D_{i}$ and $\Delta(\Drawing_{i})$ the drawing of its incident tracks. We call $S_{i}=(i,\Drawing_{i},\Delta(\Drawing_{i}))$ a state of our DP. To compute $C[S_{i}]$, we need to count all \SymbolEECrossing-crossings in $\Drawing_{i}$ involving edges from $E_{i}$, and all \SymbolTECrossing-, and \SymbolTTCrossing-crossings involving also edges or tracks incident to vertices from $V_{i-1}$ for a given state $S_{i-1}$, if the bag $V_{i-1}$ exists. Let $\CrossingCountEE{\cdot}$, $\CrossingCountTE{\cdot,\cdot}$, and $\CrossingCountTT{\cdot,\cdot}$ denote these number of crossings, respectively. Note that the exact definition of $\CrossingCountEE{\cdot}$, $\CrossingCountTE{\cdot,\cdot}$, and $\CrossingCountTT{\cdot,\cdot}$ depends on the desired drawing style. For $i>1$, the number of crossings $\CrossingCount{S_{i},S_{i-1}}$ for two states $S_{i}=(i,\Drawing_{i},\Delta(\Drawing_{i}))$ and $S_{i-1}=(i-1,\Drawing_{i-1},\Delta(\Drawing_{i-1}))$ equals the sum of $\CrossingCountEE{S_{i}}$, $\CrossingCountTE{S_{i},S_{i-1}}$, and $\CrossingCountTT{S_{i},S_{i-1}}$. We can now relate the different states in $C$ to each other as follows, where $S_{i}=(i,\Drawing_{i},\Delta(\Drawing_{i}))$, $i\in[k]$, $\Drawing_{i}\in\mathcal{G}_{i}$, and $\Delta(\Drawing_{i})\in\mathcal{D}(\Drawing_{i})$:

[TABLE]

Using induction, we can show that Equation˜1 captures the minimum number of crossings.

Lemma 4.2*.*

Let $T$ be a path decomposition of $G$ with $k$ bags. For each $i\in[k]$, drawings $\Drawing_{i}\in\mathcal{G}_{i}$ and $\Delta(\Drawing_{i})\in\mathcal{D}(\Drawing_{i})$, the table entry $C[i,\Drawing_{i},\Delta(\Drawing_{i})]$ equals the minimum number of crossings of a witness drawing for the bags $V_{1},\ldots,V_{i}$ with the drawings $\Drawing_{i}$ and $\Delta(\Drawing_{i})$ for $G_{i}$ and its incident tracks, respectively.

{prooflater}

plemmaCorrectness We use induction over $i$ to show correctness. Throughout the proof, we let $\Drawing_{i}$ and $\Delta(\Drawing_{i})$ be two arbitrary drawings.

\proofsubparagraph

*Base Case ($\boldsymbol{i=1}$). For $i=1$, we are seeking a drawing of the bag $V_{1}$ only, i.e., a drawing of $G_{1}$ in the disk $D_{1}$. Clearly, the minimum number of crossings equals the number of \SymbolEECrossing-crossings in the disk $D_{1}$, which is precisely $\CrossingCountEE{1,\Drawing_{1},\Delta(\Drawing_{1})}$; see also the first case in Equation˜1. Thus, $C[1,\Drawing_{1},\Delta(\Drawing_{1})]$ correctly captures the required crossing count.

\proofsubparagraph

*Inductive Step. Let $i>1$ and note that we are in the second case of Equation˜1. Furthermore, let $C[i,\Drawing_{i},\Delta(\Drawing_{i})]=c$ and assume, for the sake of a contradiction, that there exists a crossing-minimal witness drawing $\Drawing^{*}$ for the first $i$ bags and with the drawings $\Drawing_{i}$ and $\Delta(\Drawing_{i})$ for the disk $D_{i}$. Let the number of crossings for $\Drawing^{*}$ be some $c^{*}<c$. We now consider the drawing $\Drawing^{\prime}$ on the first $i-1$ bags that is induced by $\Drawing^{*}$ and denote by $c^{\prime}$ its number of crossings. Let $\Drawing_{i-1}$ be the drawing of $G_{i-1}$ and $\Delta(\Drawing_{i-1})$ be the drawing of its incident tracks in $\Drawing^{\prime}$. Clearly, $c^{\prime}$ is the minimum number of crossings of a witness drawing for the first $i-1$ bags of $T$ that uses the drawings $\Drawing_{i-1}$ and $\Delta(\Drawing_{i-1})$ for $G_{i-1}$ and its incident tracks, respectively. If this would not be the case, then $\Drawing^{*}$ would not be crossing-minimal. By our inductive hypothesis, we have $C[i-1,\Drawing_{i-1},\Delta(\Drawing_{i-1})]=c^{\prime}$. Note that we consider in the second case of Equation˜1 all possible states for $i-1$, and in particular the state $(i-1,\Drawing_{i-1},\Delta(\Drawing_{i-1}))$. Since we take among all such states the one that leads to a minimum number of crossings, the fact that we have $C[i,\Drawing_{i},\Delta(\Drawing_{i})]=c>c^{*}$ yields a contradiction to the existence of $\Drawing^{*}$ with $c^{*}$ crossings, as we would have set $C[i,\Drawing_{i},\Delta(\Drawing_{i})]$ to its number of crossings $c^{*}$ otherwise. Hence, by induction, the lemma statement follows.

With Lemma˜4.2 at hand, we now have all key ingredients for our DP. It remains to define for each drawing style the representations of $\Drawing_{i}$ and $\Delta(\Drawing_{i})$, if needed, and to implement the crossing counting functions $\CrossingCountEE{\cdot}$, $\CrossingCountTE{\cdot,\cdot}$, and $\CrossingCountTT{\cdot,\cdot}$. This will be the main task for the remainder of this and the upcoming sections.

Two-Page Linear Drawings.

We now aim to compute a crossing-minimal \SymbolLinear[2]-drawing of a path decomposition $T$. Recall that a two-page book drawing $\Drawing_{i}$ of $G_{i}=(V_{i},E_{i})$ is uniquely defined by the linear order $\prec_{i}$ on $V_{i}$ and the page assignment $\sigma_{i}\colon E_{i}\to\{\ell,r\}$. Thus, we set $\Drawing_{i}=\langle\prec_{i},\sigma_{i}\rangle$. As the tracks are drawn using straight lines between the vertices in the respective disks, it is sufficient to store only the drawing for each $G_{i}$, $i\in[k]$, in our DP table $C$. As $\mathcal{G}_{\max}=\mathcal{O}(\DecWidth*!\cdot 2^{\DecWidth*^{2}})$ holds, the size of the table $C$ is in $\mathcal{O}(k\cdot\DecWidth*!\cdot 2^{\DecWidth*^{2}})$ and it only remains to implement the crossing counting functions.

For \SymbolEECrossing-crossings, we observe that $\CrossingCountEE{\cdot}$ equals the number of edges with alternating endpoints in $\prec_{i}$ but assigned in $\sigma_{i}$ to the same page; see Figure˜5a and Equation˜2.

[TABLE]

To determine the number of \SymbolTECrossing-crossings, it is sufficient to observe that an edge $(u,v)\in E_{i-1}$ with $u\prec_{i-1}v$ crosses with a track for the vertex $w\in V_{i-1}\cap V_{i}$ if and only if $u\prec_{i-1}w\prec_{i-1}v$ and $\sigma_{i-1}((u,v))=r$ as visualized in Figure˜5b. A symmetric observation can be made for edges $(u,v)\in E_{i}$ with $\sigma_{i}((u,v))=\ell$, yielding the following function:

[TABLE]

This leaves us with counting the \SymbolTTCrossing-crossings, for which it is sufficient to count the number of inversions between $\prec_{i-1}$ and $\prec_{i}$; see Figure˜5c and Equation˜4.

[TABLE]

Combining this with the recurrence relation from Equation˜1, we obtain the following:

Theorem 4.3*.*

Let $T$ be a path decomposition of $G$ of width $\DecWidth$. We can compute a crossing-minimal $\SymbolLinear[2]$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{2}\cdot 4^{\DecWidth*^{2}}\cdot\DecWidth*^{4})$ time.

Proof 4.4*.*

We employ our DP algorithm storing the above-described states to compute a crossing-minimal $\SymbolLinear[2]$-drawing of $T$. Recall that the table $C$ has size $\mathcal{O}(k\cdot\DecWidth*!\cdot 2^{\DecWidth*^{2}})$, where $k=\mathcal{O}(n)$ denotes the number of bags in $T$. To compute a single state for some $i\in[k]$, we have to access up to $\mathcal{O}(\DecWidth*!\cdot 2^{\DecWidth*^{2}})$ states for $i-1$. A closer analysis of Equations˜2, 3 and 4 reveals that we can evaluate the function $\CrossingCount{\cdot,\cdot}$ in $\mathcal{O}(\DecWidth*^{4})$ time. Thus, we can compute the minimum number of crossings for a given state in $\mathcal{O}(\DecWidth*!\cdot 2^{\DecWidth*^{2}}\cdot\DecWidth*^{4})$ time. Overall, this amounts to $\mathcal{O}(k\cdot(\DecWidth*!)^{2}\cdot 4^{\DecWidth*^{2}}\cdot\DecWidth*^{4})$ time to fill the entire table $C$. The minimum number of crossings can be obtained by taking the minimum over all $C[k,\cdot,\cdot]$. We can use standard backtracking techniques to compute the crossing-minimal $\SymbolLinear[2]$-drawing of $T$. The correctness of our algorithm follows directly from the correctness of our recurrence relation, i.e., Equation˜1, established in Lemma˜4.2.

With the use of weights, we can prioritize some types of crossings over others. Furthermore, by restricting the DP table $C$ and relation from Equation˜1, we can generalize the above-presented DP. For example, to compute an \SymbolLinear[1]-drawing, we can remove the page assignment from our definition of a drawing $\Drawing_{i}$ of $G_{i}$. Furthermore, it could be desired to enforce a consistent linear order across different bags, i.e., forbid \SymbolTTCrossing-crossings. To that end, we can consider in Equation˜1 only those linear orders $\prec^{\prime}$ that are consistent with $\prec$, i.e., where $\prec\mid_{V_{i-1}}\ =\ \prec^{\prime}\mid_{V_{i}}$. Below, we summarize the effects on the running time.

Corollary 4.5*.*

Let $T$ be a path decomposition of $G$ of width $\DecWidth$. We can compute a crossing-minimal

• •

$\SymbolLinear[1]$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{2}\cdot\DecWidth*^{4})$ time.

• •

$\SymbolLinear[2]$-drawing of $T$ without \SymbolTTCrossing-crossings in $\mathcal{O}(n\cdot(\DecWidth*!)^{2}\cdot 4^{\DecWidth*^{2}}\cdot\DecWidth*^{4})$ time.

4.2 Linear Drawings for Tree Decompositions

Similarly, for graphs of bounded treewidth, we can obtain the minimum number of total crossings for $\SymbolLinear$-drawings via DP. Given a tree decomposition $T$ of $G$, we orient $T$ “left-to-right”, where the root bag is leftmost and children are to the right of the parent. We then obtain a right-to-left ordering of the tree’s bags, $V_{k},V_{k-1},\dotsc,V_{1}$, where $V_{1}$ is the root bag.

DP for Treewidth.

Our DP formulation in this problem uses the same table as for path decompositions, but a given bag may now have more than one previous bag that can affect its crossings. We assume that the tree decomposition $T$ only has bags of degree up to three. As such, any bag has at most one parent (due to the “left-to-right” orientation) and at most two children. We consider three cases to calculate $C[S_{i}]$ for a bag $V_{i}$, based on its in-degree $\deg^{-}(V_{i})\in\{0,1,2\}$. The cases where $\deg^{-}(V_{i})$ equal zero or one are similar to the two cases presented in the DP for pathwidth, but when $\deg^{-}(V_{i})=2$ the crossing function $\CrossingCount{\cdot,\cdot,\cdot}$ depends on the current state and its two child states. In this case, the current bag has two children belonging to its in-degree neighboring set $N^{-}(V_{i})$ whose embedding (which child is above the other) is decided by the DP. Our DP table has the following recurrence relation:

[TABLE]

Two-Page Linear Drawings.

The number of $\SymbolEECrossing$-crossings does not change within the parent bag, so that remains the same as in Equation˜2. The number of $\SymbolTECrossing$-crossings can be treated independently for both children $V_{x},V_{y}\in N^{-}(V_{i})$, as there are no such crossings between the children. So Equation˜6 is similar to Equation˜3 except that we sum over both children.

[TABLE]

For $\SymbolTTCrossing$-crossings, we must handle $\SymbolTTCrossing$-crossings between children. Without loss of generality, let the child $V_{x}$ that comes first in the input be above child $V_{y}$. Then such $\SymbolTTCrossing$-crossings will occur when two tracks from the children “criss-cross”, where a track from the top child goes below the track from the bottom child as in Figure˜5d, captured by Equation˜7. Then we add the $\SymbolTTCrossing$-crossings between parent and child as in Equation˜4, but for both children now:

[TABLE]

A similar analysis to Theorems˜4.3 and 4.5 yields the following results for \SymbolLinear-drawings:

Theorem 4.6*.*

Let $T$ be a tree decomposition of $G$ of width $\DecWidth$. We can compute a crossing-minimal $\SymbolLinear[2]$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{3}\cdot 8^{\DecWidth*^{2}}\cdot\DecWidth*^{4})$ time.

Proof 4.7*.*

The table $C$ has the same size $\mathcal{O}(k\cdot\DecWidth*!\cdot 2^{\DecWidth*^{2}})$, with $k=\mathcal{O}(n)$, and computing $\CrossingCount{\cdot,\cdot,\cdot}$ still takes $\mathcal{O}(\DecWidth*^{4})$ time as in the pathwidth case. However, computing a single state must now check all possible pairings of its two child states, yielding $\mathcal{O}((\DecWidth*!\cdot 2^{\DecWidth*^{2}})^{2})$ states in the worst case. So, altogether, the running time will be in $\mathcal{O}(n\cdot(\DecWidth*!)^{3}\cdot 8^{\DecWidth*^{2}}\cdot\DecWidth*^{4})$.

Corollary 4.8*.*

Let $T$ be a tree decomposition of $G$ of width $\DecWidth$. We can compute a crossing-minimal $\SymbolLinear[1]$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{3}\cdot\DecWidth*^{4})$.

Finally, we note that our DP algorithm can be extended to decompositions $T$ with bags of arbitrary degree by generalizing Equations˜5, 6 and 7. In particular, for Equation˜7, we must account for \SymbolTTCrossing-crossings between every pair of children of $V_{i}$. As stated in Equation˜5, we consider all possible combinations of states for $V_{i}$’s children, and the number of such combinations grows with their number. Moreover, observe that the embedding of these children affects the number of \SymbolTTCrossing-crossings, and the number of potential embeddings is itself exponential in the degree $\Delta$ of $V_{i}$. Therefore, allowing bags with non-constant degree in $T$ introduces an exponential factor in terms of $\Delta$ in the running time of our DP algorithm.

5 Drawing Styles with Circular Vertex Placements

In this section, we discuss how to adapt our DP algorithm presented in Section˜4 to compute crossing-minimal \SymbolCircular- and \SymbolOrbital-drawings.

Circular Drawings.

Circular drawings arrange the vertices in every disk $D_{i}$, $i\in[k]$, on a circle and draw the edges $E_{i}$ as straight lines. Thus, the drawing $\Drawing_{i}$ of $G_{i}$ is uniquely defined by the placement of the vertices $V_{i}$ in $D_{i}$. Since they are evenly distributed on a circle, it suffices to store in $\Drawing_{i}$ for each bag $V_{i}\in T$, the counterclockwise order $\prec_{i}$ of the vertices $V_{i}$ as they appear in $D_{i}$, starting at the twelve o’clock position. Recall that the angle parameter $\alpha$ specifies the starting angle of the first vertex in the order $\prec_{i}$ in the disk $D_{i}$. As the tracks are again straight lines, we do not need to store $\Delta(\Drawing_{i})$, reducing the size of $C$ to $\mathcal{O}(k\cdot\DecWidth*!)$.

When filling the table $C$, we recall that the recurrence relations from Equations˜1 and 5 include the number of \SymbolEECrossing-, \SymbolTECrossing-, and \SymbolTTCrossing-crossings involving the vertices of a bag $V_{i}$, $i\in[k]$, and those of its adjacent bags (if they exist). The number of \SymbolEECrossing-crossings corresponds, due to the equivalence with one-page book drawings, to the number of edge pairs with alternating endpoints in $\prec_{i}$ and is, therefore, purely combinatorial in this setting. In contrast, the \SymbolTECrossing-, and \SymbolTTCrossing-crossings are geometric in nature and depend on the concrete positions of the vertices within the disks. Observe that these positions depend on $\alpha$ and we remark that the choice of $\alpha$ influences the number of observed crossings; see also Figure˜6. Consequently, we equip the crossing counting functions $\CrossingCountTE{\cdot,\cdot,\cdot}$ and $\CrossingCountTT{\cdot,\cdot,\cdot}$ with the parameter $\alpha$. Altogether, this is sufficient to compute crossing-minimal \SymbolCircular-drawings.

Orbital Drawings.

Orbital drawings extend circular drawings by routing the tracks as orbits around the vertices; recall Figure˜2d. Since tracks must remain within the track-routing area, every \SymbolOrbital-drawing is free of \SymbolTECrossing-crossings. However, unlike in the other two drawing styles, the drawing of tracks within a disk is no longer uniquely determined by the placement of the vertices. In particular, tracks can be assigned to different orbits and may orbit the drawing $\Drawing_{i}$ of $G_{i}$ either clockwise or counterclockwise. Thus, while it still suffices to store a linear order $\prec_{i}$ of $V_{i}$ for $\Drawing_{i}$, representing $\Delta(\Drawing_{i})$ now requires additional care.

We model $\Delta(\Drawing_{i})$ with two parts. First, to model the assignment of tracks to orbits, we introduce an orbit assignment function $\lambda_{i}\colon V_{i}\to[\DecWidth*]$, where $\lambda_{i}(v)$ specifies the orbit used for the tracks of vertex $v$, numbered from the center of the disk $D_{i}$ outward; see Figure˜7. Two tracks for different vertices $u,v\in V_{i}$ may share the same orbit if they do not intersect. Otherwise, we require $\lambda_{i}(u)\neq\lambda_{i}(v)$ to guarantee that no two tracks for different vertices share the same orbit simultaneously. Second, to model the direction in which the tracks orbit, i.e., clockwise or counterclockwise, we consider the (up to two) children $V_{x}$ and $V_{y}$ and the parent $V_{q}$ of the bag $V_{i}$ in the tree decomposition $T$, if they exist. We define a direction function $\delta_{i}\colon V_{i}\times\{x,y,q\}\to\{\text{cw},\text{ccw}\}$ which, for each vertex $v\in V_{i}$, captures the direction (clockwise (cw) or counterclockwise (ccw)) of the track from $V_{i}$ to each adjacent bag. Figure˜7 illustrates the semantic of $\delta_{i}$ and underlines the importance of storing this information separately for each adjacent bag. Thus, for a given drawing $\Drawing_{i}$ of $G_{i}$, we set $\Delta(\Drawing_{i})=(\lambda_{i},\delta_{i})$. This increases the size of the DP table $C$ to $\mathcal{O}(k\cdot\DecWidth*!\cdot{\left(8\DecWidth*\right)}^{\DecWidth*})$.

Regarding the computation of the number of crossings, we observe that the function $\CrossingCountEE{\cdot}$ from \SymbolCircular-drawings can be reused, and that \SymbolTECrossing-crossings do not occur in this drawing style. \SymbolTTCrossing-crossings involving a track for a vertex $v\in V_{i}$ occur in the following two cases. First, if $v$ also appears in an adjacent bag, and a track for some vertex $u\in V_{i}$ with $\lambda_{i}(u)<\lambda_{i}(v)$ passes by the position of $v$ in $\Drawing_{i}$, then these two tracks cross: The track for $v$ must cross the one of $u$ to reach its orbit; see Figure˜8a. Second, for each adjacent bag $V_{j}$ with $v\in V_{j}$, we count the number of vertices $u\in V_{i}$ such that a track for $u$ blocks the path from $D_{i}$ to $D_{j}$, see Figure˜8b. In both cases, the crossings can be determined from the starting angle $\alpha$, the order of the children, and the stored information, i.e., $\Drawing_{i}$ and $\Delta(\Drawing_{i})$, for the disk $D_{i}$ of $V_{i}$ and the disks of its adjacent bags. Hence, the function $\CrossingCountTT{\cdot,\cdot,\cdot}$ still only depends on the states for $V_{i}$ and its child bags. Finally, since tracks to occurrences of the same vertex $v\in V_{i}$ in different bags can overlap, we must avoid double-counting in such cases; see again Figure˜8a.

We summarize below the findings of this section. Recall that $k=\mathcal{O}(n)$ holds.

Theorem 5.1*.*

Let $T$ be a decomposition of $G$ of width $\DecWidth$. We can compute a crossing-minimal

• •

$\SymbolCircular$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{2}\cdot\DecWidth*^{4})$ time if $T$ is a path decomposition.

• •

$\SymbolCircular$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{3}\cdot\DecWidth*^{4})$ time if $T$ is a tree decomposition.

• •

$\SymbolOrbital$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{2}\cdot{\left(4\DecWidth*\right)}^{2\DecWidth*}\cdot\DecWidth*^{4})$ time if $T$ is a path decomposition.

• •

$\SymbolOrbital$-drawing of $T$ in $\mathcal{O}(n\cdot(\DecWidth*!)^{3}\cdot{\left(8\DecWidth*\right)}^{3\DecWidth*}\cdot\DecWidth*^{4})$ time if $T$ is a tree decomposition.

{prooflater}

ptheoremCircularDrawings Let $k$ denote the number of bags of $T$. Recall $k=\mathcal{O}(n)$ and that our dynamic program (DP) traverses the decomposition $T$ and considers every bag $V_{i}\in T$, and all the possible drawings $\Drawing_{i}$ and $\Delta(\Drawing_{i})$ of $G_{i}$ and its incident tracks, respectively. For each of them, all possible combinations with states of the two children of $V_{i}$, if they exist, or $V_{i-1}$, in case $T$ is a path decomposition. Lemma˜4.2 and the arguments in Section˜4 ensure correctness of this DP for our definitions of states in the DP. Therefore, we focus for the remainder of the proof on the claimed running times and discuss $\SymbolCircular$- and $\SymbolOrbital$-drawings separately.

\proofsubparagraph

*$\SymbolCircular$-drawings. Recall that a drawing $\Drawing_{i}$ for $G_{i}$, $i\in[k]$ can, thanks to the angle parameter $\alpha$, be defined by the counterclockwise order $\prec_{i}$ of the vertices $V_{i}$ starting at the twelve o’clock position. As the tracks are straight lines, the size of the DP-table $C$ is $\mathcal{O}(k\cdot\DecWidth*!)$. The position for each vertex can be computed in $\mathcal{O}(1)$ time. Thus, the evaluation of the crossing counting function $\CrossingCount{\cdot,\cdot}$ still requires $\mathcal{O}({\DecWidth*}^{4})$ time as for \SymbolLinear-drawings. For computing one state of the DP, we must check all $\mathcal{O}(\left(\DecWidth*!\right)^{2})$ combinations of the up to two children states. This yields the claimed running for tree decompositions. For path decompositions, only $\mathcal{O}(\DecWidth*!)$ states need to be evaluated as each bag has degree at most two.

\proofsubparagraph

*$\SymbolOrbital$-drawings. Recall that for \SymbolOrbital-drawings, we need to specify in addition the drawing of the tracks inside the disk $D_{i}$, $i\in[k]$, using an orbit assignment function $\lambda_{i}$ and a direction function $\delta_{i}$. As $\mathcal{G}_{\max}=\mathcal{O}(\DecWidth*!)$ and $\mathcal{D}_{\max}=\mathcal{O}(\DecWidth*^{\DecWidth*}\cdot 2^{3\DecWidth*})$, this results in a DP-table $C$ of size $\mathcal{O}(k\cdot\DecWidth*!\cdot{\left(8\DecWidth*\right)}^{\DecWidth*})$. As for \SymbolCircular-drawings, we can compute the position for each vertex in $\mathcal{O}(1)$ time, which yields the drawing $\Drawing_{i}$ of $G_{i}$. Together with $\Delta(\Drawing_{i})=(\lambda_{i},\delta_{i})$, this is sufficient to describe the drawing of the tracks in the disk $D_{i}$. Despite not having \SymbolTECrossing-crossings, computing the number $\CrossingCount{\cdot,\cdot}$ of crossing still takes $\mathcal{O}({\DecWidth*}^{4})$ time. Since we have to evaluate for each state all combinations with the $\mathcal{O}(\DecWidth*!\cdot{\left(8\DecWidth*\right)}^{\DecWidth*})$-many states for each of its up to two children, the claimed running time for tree decompositions follows. If $T$ is a path decomposition, the fact that we have degree at most two reduces the size of the table $C$ to $\mathcal{O}(k\cdot\DecWidth*!\cdot{\left(4\DecWidth*\right)}^{\DecWidth*})$ and the evaluation time for a single state to $\mathcal{O}(\left(\DecWidth*!\right)\cdot{\left(4\DecWidth*\right)}^{\DecWidth*}\cdot\DecWidth*^{4})$.

6 Heuristic Approaches to Compute Linear Witness Drawings

A proof-of-concept implementation of our DP from Section˜4.2 for \SymbolLinear[2]-drawings of tree decompositions required over 8 minutes to terminate even for small decompositions of width $\DecWidth=3$ with four bags. This is no surprise given the running time bounds established in Theorem˜4.6. For tree decompositions of larger width, this became too large in practice with a single state needing to check up to $(5!\cdot 2^{5^{2}})^{2}\approx 10^{19}$ possible configurations for $\DecWidth=4$.

Consequently, we do not want to restrict our attention to exact, but slow, algorithms, but also explore heuristics to efficiently compute drawings with a small number of crossings, but without any formal guarantee on optimality. In this section, we describe three heuristics for \SymbolLinear[2]-drawings. We evaluate their performance in Section˜7 and provide in Figures˜9 and 3 sample drawings computed by them.

Global Book Drawing Heuristic.

Since we visualize in an \SymbolLinear[2]-drawing each graph $G_{i}$, $i\in k$, as a two-page book drawing of $G_{i}$, our first heuristic computes such a drawing $\langle\prec,\sigma\rangle$ for the entire graph $G$. After that, we set for each disk $D_{i}$ $\prec_{i}\ =\ \prec\mid_{V_{i}}$ and $\sigma_{i}(e)=\sigma(e)$ for every edge $e\in E_{i}$, i.e., we project the drawing $\langle\prec,\sigma\rangle$ down into each bag. If $T$ is a tree decomposition, we additionally perform a bottom-up traversal of $T$, choosing for each internal bag $V_{i}$ the embedding of its children that yields the fewer \SymbolTECrossing- and \SymbolTTCrossing-crossings locally.

Since computing a crossing-minimal one-page book drawing for a graph is an \NP-hard problem [44], we already employ a heuristic for this step. Klawitter, Mchedlidze, and Nöllenburg [36] performed an extensive experimentally evaluation on book drawing heuristics. In their conclusion, they recommended the heuristic conGreedy+, which places one vertex after the other on the spine, selecting the next vertex based on the number of already placed neighbors, and extends the spine order and page assignment greedily. While for two-page book drawings of $k$-trees, which are maximal graphs of treewidth $k$, it was slightly outperformed by other heuristics, it performed best overall and in particular for graphs with a sub-quadratic number of edges. The heuristic has a running time of $\mathcal{O}(m^{2}n)$ [36] and we call it Global conGreedy+.

Local Book Drawing Heuristic.

Computing a book drawing for the whole graph $G$ and then applying it to each disk $D_{i}$, $i\in k$, has two main disadvantages. On the one hand, we usually do not visualize the entire graph $G$ in a single bag, but several induced subgraphs $G_{i}$ separately. Therefore, the drawing $\langle\prec_{i},\sigma_{i}\rangle$ for $G_{i}$ might contain unnecessary crossings. On the other hand, the above-described heuristic is, apart from the embedding step, unaware of \SymbolTECrossing- and \SymbolTTCrossing-crossings. This, in particular, leads to situations where swapping the page assignment of one edge $e\in E_{i}$ or the entire edge set $E_{i}$ can dramatically reduce the number of \SymbolTECrossing-crossings. The latter occurs especially at the root and leave bags of $T$.

Therefore, we now employ the algorithm conGreedy+ in each bag $V_{i}$ separately during a top-down traversal of $T$, thus computing for each $G_{i}$ a book drawing from scratch. Since the drawing for the parent of a bag $V_{i}$ has been determined, we can take the potential \SymbolTTCrossing- and \SymbolTECrossing-crossings into account. More precisely, when selecting the spine position of a vertex $v\in V_{i}$ and the page assignment for its incident edges in $G_{i}$ we also compute the number of \SymbolTECrossing- and \SymbolTTCrossing-crossings that this causes based on the drawing of the parent, similar to Equations˜6 and 7. Furthermore, we forward this information to its children when considering their two possible embeddings if necessary. We call this heuristic Local conGreedy+ and, since the size of each $G_{i}$ is bounded, its running time is $\mathcal{O}(n\cdot\DecWidth*^{5})$.

Local Search.

In addition to the two above-mentioned heuristics, we also perform a local search to reduce the number of crossings. Given an \SymbolLinear[2]-drawing of $T$, we traverse $T$ bottom up. For each bag $V_{i}\in T$, we perform one or several of the following movements.

Vertex-Swap:

Take two vertices $u,v\in V_{i}$ and swap their position on the spine $\prec_{i}$.

Edge-Swap:

Take two edges $e_{1},e_{2}\in E_{i}$ with $\sigma_{i}(e_{1})\neq\sigma_{i}(e_{2})$ and swap their page assignment.

Edge-Flip:

Take an edge $e\in E_{i}$ and assign it to the other page.

Embedding-Flip:

Flip the embedding of the subtree $V_{i}$, i.e., the order of its children.

For a given bag $V_{i}\in T$, we perform a hill-climbing approach and exhaustively apply above movements until no further improvement can be made in the drawing $\langle\prec_{i},\sigma_{i}\rangle$. Then, we proceed to the next bag $V_{j}\in T$. Note that when evaluating the number of crossings, we keep the drawings in the remaining bags fixed. Thus, after a bottom-up traversal of $T$, we perform a second traversal, this time top-down. Afterwards, or after a predetermined period of time, we return the best solution found so far.

7 Experimental Evaluation

We implemented the dynamic program for \SymbolLinear[2]-crossings as described in Section˜4.2 as well as all of the heuristics presented in Section˜6 in Python 3.6.9. All algorithms run on a system with Intel Xeon E5-2640 v4 10-core processors at 2.40 GHz. We set hard limits for memory and time of 96 GB and fifteen minutes, respectively, but memory was not a limitation for our algorithms. If we improve a solution from a heuristic using local search, we indicate this using “& LS”. Also the local search has a maximum running time of fifteen minutes. The computed drawings are visualized using d3.js on an independent system and this final step is not considered here. The instances are based on a public repository containing graphs extracted from “Sage” together with a tree decomposition of them.111The original dataset is available at https://github.com/freetdi/named-graphs. The filtered dataset, source code, and evaluation code can be found on OSF [16]. The application can also be accessed online at https://www.ac.tuwien.ac.at/projects/visualizing-treewidth/.

Out of the 150 graphs, 55 of them were equipped with a (tree or path) decomposition $T$ where each bag, i.e., node of the decomposition $T$, has degree at most three in $T$ and we used these in our experiments. We have no information on how these decompositions were computed or if they are optimal. The width of the decompositions ranges between one and 53 and they have between one and 143 bags. For 60% of the instances, both parameters were simultaneously below ten. The graphs have between four and 143 vertices and between six and 1,008 edges, with a mean density, i.e., $2m/(n(n-1))$, of $0.37$.

Computed Drawings.

We provide in Figures˜9 and 3 sample drawings of two graphs together with the witness drawings computed by our different algorithms. Generally, we can see that the DP and Global conGreedy+ produce drawings with a more consistent arrangement of the vertices on the individual spines. Comparing Figures 9b and f, we can observe that the obtained drawing from Global conGreedy+ & LS is nearly identical to the one of the DP, confirming the observed low crossing numbers. The consistency across different spines yields also a visually more pleasing drawing, as we can see in Footnote˜3. We provide in \shortLongthe full version [ARXIV]Appendix˜A more sample drawings computed by our algorithms and analyze below their running time and the quality of the produced drawings.

Running Times, Tradeoffs, and the Influence of the Width.

All heuristics terminated on every instance within fifteen minutes. The DP could solve only thirteen instances within the time limit. To evaluate the crossing minimization performance of the heuristics on a larger set of instances, we ran the DP with a time limit of six hours, after which 21 of the 55 instances, all of width at most four, could be solved optimally. For the remaining 34 instances, Global conGreedy+ obtained four, Local conGreedy+ three, Global conGreedy+ & LS 18, and Local conGreedy+ & LS 20 times a solution with the fewest crossings.

Figure˜11 shows the relative number of crossings, compared to the best-performing algorithm, i.e., the DP where available and one of the heuristics otherwise, and running time for each algorithm and instance. We group instances by width and sort them within each group based on the number of bags. In \shortLongthe full version [ARXIV]Figure˜12, we provide a filtered version of Figure˜11, focusing on instances for which the DP terminated within six hours. Regarding the running time, we can observe in Figure˜11a that all heuristics without the local search step terminated within a second on every instance except the largest ones.

As expected, Global conGreedy+ is the fasted heuristic, since it has to compute only one book drawing for the entire graph. Regarding the optimality-ratio, Figure˜11b shows that for instances with larger width, Global conGreedy+ seems to work slightly better than Local conGreedy+. We believe that computing a book drawing for each $G_{i}$ individually could yield situations where the obtained spine orders yield many \SymbolTTCrossing-crossings. In particular, we suspect that spine orders that “locally” admit few crossings could “globally” introduce, for example, unexpected “criss-cross” \SymbolTTCrossing-crossings (recall Figure˜5d). We see the strength of our DP for decompositions of width two or three. For larger width, the high running time becomes impractical. However, small-width instances are often simple enough that all heuristics can solve them to optimality in a few milliseconds. Interestingly, for instances with very large treewidth, the benefit of local search drastically decreases: Despite performing movement operations for the entire fifteen minutes, the obtained solution is, if at all, only negligibly better then the corresponding heuristic without the local search. However, for instances of moderate width, local search considerably improved the initial solution.

{statelater}

perInstancePlotDP

8 Concluding Remarks and Future Directions

In this paper, we have initiated the study of visualizing tree and path decompositions of graphs to visually certify their treewidth and pathwidth, respectively. Our experimental study shows that for decompositions of width at most three, we can compute exact crossing-minimal drawings in a few seconds. For decompositions of medium to larger width, the heuristics combined with a local search step seem to be the preferred method. We see the computation of a crossing-minimal drawing where we do not restrict the placement of the child bags or the starting angle inside the disk as a challenging open algorithmic problem. Furthermore, the taxonomy presented Section˜3 can be seen as a first step towards exploring the design space for witness drawings of graphs with bounded pathwidth or treewidth. In particular, future work should explore further drawing styles for (i) the graph $G_{i}$ inside each bag $V_{i}$, (ii) tracks between the bags, (iii) the underlying tree (or path) $T$, and possibilities (iv) to distinguish different vertices other than color. On top of that, a comprehensive design study can also consider (v) the option to not visualize an edge $uv$ in every bag $V_{i}$ with $u,v\in V_{i}$ but only in one such bag, (vi) the algorithmic and visual impact of the structure of the underlying tree/path $T$, e.g., its number of bags, their maximum degree, or the “niceness” of $T$, and (vii) further optimization criteria other than the (unweighted) number of crossings.

Additionally, we believe that the exploration of the design space should go hand in hand with the development of further exact and heuristic algorithms and an empirical evaluation of the drawing styles proposed in this article or extracted from the (extended) taxonomy, e.g., with a user study. Finally, designing and computing suitable witness drawings for other width parameters of graphs is an interesting, independent direction for future work.

Appendix A Sample Drawings

We provide in Figures˜13 and 14 further witness drawings computed by our algorithms. Figure˜13 shows four drawings computed by the DP. Figure˜14 shows the drawings for four decompositions computed by Global conGreedy+ & LS and Local conGreedy+ & LS. We provide for all drawings the number of crossings in the parenthesis. For further information on the respective graphs, we refer to the overview page of Sage444https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_generators.html, on which also the dataset is based on.