
TL;DR
This paper introduces the concept of planar order on vertex posets of processive planar graphs, establishing a natural correspondence with the planar order on their edge posets, and extends the concept to general finite posets.
Contribution
It proves the existence of a natural planar order on vertex posets derived from the edge poset's planar order in processive planar graphs.
Findings
Planar order on vertex posets is equivalent to conjugate order.
A natural planar order on vertex posets can be induced from edge posets.
The concept extends to any finite poset, not just graphs.
Abstract
A planar order is a special linear extension of the edge poset (partially ordered set) of a processive plane graph. The definition of a planar order makes sense for any finite poset and is equivalent to the one of a conjugate order. Here it was proved that there is a planar order on the vertex poset of a processive planar graph naturally induced from the planar order of its edge poset.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · graph theory and CDMA systems
Planar order on vertex poset
Xuexing Lu
School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, Hefei, China
Abstract
A planar order is a special linear extension of the edge poset (partially ordered set) of a processive plane graph. The definition of a planar order makes sense for any finite poset and is equivalent to the one of a conjugate order. Here it was proved that there is a planar order on the vertex poset of a processive planar graph naturally induced from the planar order of its edge poset.
Keywords: edge poset, vertex poset, planar order
1 Introduction
The notion of a processive plane graph, a special case of Joyal and Street’s progressive plane graph [2], was introduce in [3] as a graphical tool for tensor calculus in semi-groupal categories. In [3], we gave a totally combinatorial characterization of an equivalence class of processive graphs in terms of the notions of a POP-graph which is a processive graph (a special kind of acyclic directed graph) equipped with a planar order (a special linear order of the edges).
However, it turns out that the notion of a planar order can be defined for a general finite poset (partially ordered set) and essentially equivalent to the one of a conjugate order [1], which is an important notion in the study of planar posets. So this raises an interesting question: for a processive graph, are there some relations between planar orders on its edges and planar orders on its vertices? In this paper, we will give a positive answer to this question by showing that any planar order of edges of a processive graph naturally induces a planar order of vertices.
2 Processive plane graph
Definition 1**.**
A processive plane graph is an acyclic directed graph drawn in a plane box with the properties that: all edges monotonically decrease in the vertical direction; all sources and sinks are of degree one; and all sources and sinks are placed on the horizontal boundaries of the plane box.
Figure shows an example.
Figure . A processive plane graph
Processive plane graphs can also be defined in terms of processive graphs [3] and their boxed planar drawings.
Definition 2**.**
A processive graph is an acyclic directed graph with all its sinks and sources of degree one.
A planar drawing of processive graph is called boxed [2] if is drawn in a plane box with all sinks of on one horizontal boundary of the plane box and all sources of on the other horizontal boundary of the plane box. A planar drawing of an acyclic directed graph is called upward if all edges increases monotonically in the vertical direction (or other fixed direction). Thus a processive plane graph is exactly a boxed and upward planar drawing of a processive graph.
Definition 3**.**
Two processive plane graphs are equivalent if they are connected by a planar isotopy such that each intermediate planar drawing is boxed (not necessarily upward).
In [3], equivalence classes of processive plane graphs are mainly used to construct a free strict tensor category on a semi-tensor scheme.
3 Planar order and POP-graph
In [3], we gave a combinatorial characterization of an equivalence classes of a processive plane graph in terms of a planar order on its underlying processive graph. In this paper, we define planar order for any poset.
Definition 4**.**
A planar order on a poset is a linear order on , such that
* for any , implies ;*
* for any , and imply that either or .*
says that is a linear extension of .
Recall that two partial orders on a set are conjugate if each pair of elements are comparable by exactly one of them. It is easy to see that is equivalent to the condition that if , then and imply that . Thus enables us to define a transitive binary relation: if and only if and ; moreover, if is satisfied, then the linearity of implies that is a conjugate order of . So the planar order is a reformulation of the conjugate order of .
In a directed graph, we denote if there is a directed path starting from edge and ending with edge . Similarly, denotes that there is a directed path starting from vertex and ending with vertex . For any acyclic directed graph, its edge set and vertex set are posets with the relation and . We call them edge poset and vertex poset of the acyclic directed graph, respectively.
The following is a key notion in [3].
Definition 5**.**
A planarly ordered processive graph or POP-graph , is a processive graph together with a planar order on its edge poset .
We simply denote a POP-graph as ; see Fig for an example.
6$$5$$9$$12$$15$$8$$19$$1$$2$$3$$4$$10$$11$$17$$7$$13$$14$$16$$18
Figure . A POP-graph
A basic result is the following.
Theorem 6** ([3]).**
There is a bijection between POP-graphs and equivalence classes of processive plane graphs.
The POP-graph in Fig corresponds to the processive plane graph in Fig .
4 Planar order on vertices
In this section, we will prove our main result. Before that we need some preliminaries.
From now on, we fix be a POP-graph . For a vertex of , the set of incoming edges and the set of outgoing edges are linearly ordered by . We introduce some notations when or are not empty:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The following lemma is a result first proved in [4].
Lemma 7**.**
Let be a vertex of . If the degree of is not one, then under the linear order .
Proof.
Notice that is a processive graph, then implies that and . Thus both and exist. Now we prove by contradiction. Suppose there exists an edge , such that . Since , then by we have or . If , then there must exists an edge , such that or . Thus , which contradicts . Otherwise, , then there must exist an edge such that or . Then , which contradicts . ∎
Lemma 7 shows that for any vertex , , where is the set of incident edges of and denotes the interval of subset in a poset. Due to Lemma 7, we can define a linear order on the vertex set . For any two different vertices of , if and only if one of the following conditions is satisfied:
, , , .
We write if or . The following Theorem is our main result.
Theorem 8**.**
For any POP-graph , defines a planar order on the vertex poset .
Proof.
satisfies . If , then there exist such that , and for , which implies that . Thus , then by definition of , we have .
satisfies . Suppose and , then and exist and . We have four cases:
Case 1: is a source and is a sink. In this case, notice that is processive, then by Definition , and . So implies that . Let or , then implies that or or . In the first case, by , we have or , which implies that or . In the second case, we have , and in the third case, we have .
Case 2: is not a source and is a sink. In this case, exists and by Definition , . So implies that . Let or , then implies that or . In the first case, by , we have or , which implies that or . In the second case, we have .
Case 3: is a source and is not a sink. This case is similar to case .
Case 4: is not a source and is not a sink. In this case, both and exist and implies that . Let or , then implies that . By , we have or , which implies that or . ∎
Figure shows the planar order on the vertex poset of the POP-graph in Fig .
2$$3$$4$$1$$7$$12$$13$$15$$8$$9$$16$$18$$5$$6$$11$$10$$14$$17$$19$$20
Figure . Induced planar order on vertices.
Theorem 8 shows that for any processive graph, each conjugate order of its edge poset induces a conjugate order of its vertex poset. However, in general, the converse is not true. Therefore, together with Theorem 2.1, Theorem 3.1 demonstrates that edge poset is more effective tool than vertex poset in the study of upward planarity. It is worth to mention that Fraysseix and Mendez, in a different but essentially equivalent context, also showed a similar judgement (in their final remark of [1]). In our subsequent work, we will show that for a transitive reduced processive graph (directed covering graph of a poset), a planar order on its vertex set can naturally induce a planar order on its edge set, which is essentially related the work in [1].
Acknowlegement
This work is supported by "the Fundamental Research Funds for the Central Universities".
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] H.de Fraysseix and P.O. de Mendez. Planarity and edge poset dimension. European J. Combin. , 17(8):731-740, 1996.
- 2[2] A. Joyal and R. Street. The geometry of tensor calculus. I. Adv. Math. , 88(1):55–112, 1991.
- 3[3] S. Hu, X. Lu and Y. Ye. A graphical calculus for semi-groupal categories. Applied Categorical Structures , 27(2):163–197, April 2019, ar Xiv:1604.07276.
- 4[4] X. Lu and Y. Ye. Combinatorical characterization of upward planarity. Communications in Mathematics and Statistics , 7(2):207–223, June 2019, ar Xiv:1608.07255.
