The structure of k-planar graphs

07/11/2019 ∙ by Vida Dujmovic, et al. ∙ 0

Dujmović et al. (FOCS 2019) recently proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path. This tool has been used to solve longstanding problems on queue layouts, non-repetitive colouring, p-centered colouring, and implicit graph encoding. We generalise this result for k-planar graphs, where a graph is k-planar if it has a drawing in the plane in which each edge is involved in at most k crossings. In particular, we prove that every k-planar graph is a subgraph of the strong product of a graph of treewidth O(k^5) and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that k-planar graphs have non-repetitive chromatic number upper-bounded by a function of k. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a much more general setting based on so-called shortcut systems that are of independent interest.

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1 Introduction

A graph is -planar if it has a drawing in the plane in which each edge is involved in at most crossings. Such graphs provide a natural generalisation of planar graphs, and are important in graph drawing research; see the recent bibliography on 1-planar graphs and the 140 references therein [25]. The present paper studies the structure of -planar graphs and other more general classes of graphs. The main results of this paper generalise the following recent theorem of Dujmović et al. [15] to a setting that includes -planar graphs.

[[15]] Every planar graph is a subgraph of , for some graph of treewidth at most and for some path .

Here is the strong product, and treewidth is an invariant that measures how ‘tree-like’ a given graph is; see Section 2 for formal definitions and see Figure 1 for an example. Loosely speaking, Section 1 says that every planar graph is contained in the product of a tree-like graph and a path. This enables combinatorial results for graphs of bounded treewidth to be generalised for planar graphs (with different constants). Section 1 has been the key tool in solving several well-known open problems. In particular, Dujmović et al. [15] use it to prove that planar graphs have bounded queue-number (resolving a conjecture of Heath et al. [21] from 1992); Dujmović et al. [14] use it to prove that planar graphs have bounded non-repetitive chromatic number (resolving a conjecture of Alon et al. [4] from 2002); and Bonamy et al. [6] use it to find shorter implicit representations of planar graphs (making progress on a sequence of results going back to 1988 [23, 24]).

Figure 1: Example of a strong product.

We generalise Section 1 as follows.

Every -planar graph is a subgraph of , for some graph of treewidth and for some path .

Although -planar graphs are the most high-profile target for a generalization of Section 1, we actually prove a substantially stronger result than Section 1 using the following definition. A collection of paths in a graph is a -shortcut system (for ) if:

  • every path in has length at most ,111A path of length consists of edges and vertices. and

  • for every , the number of paths in that use as an internal vertex is at most .

Each path is called a shortcut; if has endpoints and then it is a -shortcut. Given a graph and a -shortcut system for , let denote the supergraph of obtained by adding the edge for each -shortcut in .

This definition is related to -planarity because of the following observation:

Every -planar graph is a subgraph of for some planar graph and some -shortcut system for .

The proof of Section 1 is trivial: Given a -plane embedding of a graph , create a planar graph by adding a dummy vertex at each crossing point. For each edge there is a path in between and of length at most . Let be the set of such paths . For each vertex of , at most two paths in use as an internal vertex (since no original vertex of is an internal vertex of a path in ). Thus is a -shortcut system for , such that .

We prove the following theorem that shows if a graph is a subgraph of and is a shortcut system for , then is a subgraph of , where the treewidth of is bounded by a function of the treewidth of .

Let be a subgraph of , for some graph of treewidth at most and for some path . Let be a -shortcut system for . Then is a subgraph of for some graph of treewidth at most and some path .

Sections 1, 1 and 1 immediately imply Section 1 with instead of . Some further observations reduce the bound to ; see Section 2.

Section 1 leads to several other results of interest. Here is one example. The -th power of a graph is the graph with vertex set , where if and only if .222For a graph and two vertices , denotes the length of a shortest path, in , with endpoints and . We define if and are in different connected components of . If has maximum degree , then for some -shortcut system ; see Section 8.3. Sections 1 and 1 then imply:

For every planar graph with maximum degree and for every integer , is a subgraph of , for some graph of treewidth at most .

These theorems have applications in diverse areas, including queue layouts [15], non-repetitive colouring [14], and -centered colouring [10], which we explore in Section 6. For example, we prove that -planar graphs have bounded non-repetitive chromatic number (for fixed ). Prior to the recent work of Dujmović et al. [14], it was even open whether planar graphs have bounded non-repetitive chromatic number.

Section 8 presents several examples of graph classes that can be obtained from a shortcut system applied to a planar graph, including graph powers, map graphs, string graphs, and nearest neighbour graphs. All of these results also apply, where instead of planar graphs, we consider graphs of bounded Euler genus. All of the applications discussed in Section 6 work on these graph classes.

2 Layerings, Decompositions and Partitions

This section defines concepts and results from the literature that will be important for our work.

In this paper, all graphs are finite and undirected. Unless specifically mentioned otherwise, all graphs are also simple. For any graph and any set (typically ), let denote the graph with vertex set and edge set . We use as a shorthand for . We use to denote subgraph containment; that is, and .

We now formally define -planar graphs. An embedded graph is a graph with in which each edge is a closed curve333A closed curve in a surface is a continuous function . The points and are called the endpoints of the curve. When there is no danger of misunderstanding we treat a curve as the point set . in with endpoints and and not containing any vertex of in its interior. A crossing in an embedded graph is a triple with , and such that . An embedded graph is -plane if each edge of takes part in at most crossings. A (not necessarily embedded) graph is -planar if there exists a -plane graph isomorphic to . Under these definitions, -planar graphs are exactly planar graphs and -plane graphs are exactly plane graphs.

We now define two concepts used in the theorems in Section 1: strong products and treewidth. The strong product of graphs and , denoted by , is the graph with vertex set , where distinct vertices are adjacent if:

  • and , or

  • and , or

  • and .

A tree-decomposition of a graph consists of a tree and a collection of subsets of indexed by the nodes of such that:

  1. for every , there exists some node with ; and

  2. for every , the induced subgraph is connected.

The width of the tree-decomposition is . The treewidth of a graph is the minimum width of a tree-decomposition of . Treewidth is the standard measure of how similar a graph is to a tree. Indeed, a connected graph has treewidth 1 if and only if it is a tree. Treewidth is of fundamental importance in structural and algorithmic graph theory; see [31, 20, 5] for surveys.

While strong products enable concise statements of the theorems in Section 1, to prove such results it is helpful to work with layerings and partitions, which we now introduce.

A layering of a graph is a sequence such that is a partition of and for every edge , if and then . For any partition of , a quotient graph has a -element vertex set and if and only if there exists an edge such that and . To highlight the importance of the quotient graph , we call an -partition and write this concisely as so that each element of is indexed by the vertex it creates in .

For any partition of and any layering of we define the layered width of with respect to as . For any partition of , we define the layered width of as the minimum, over all layerings of , of the layered width of with respect to .

2.1 Previous Work on Partitions of Minor-Closed Classes

Dujmović et al. [15] introduced the study of partitions with bounded layered width such that the quotient has some additional desirable property, like small treewidth. They defined a class of graphs to admit bounded layered partitions if there exist such that every graph has an -partition of layered width at most for some graph of treewidth at most .

These definitions relate to strong products as follows.

[[15]] For every graph , a graph has an -partition of layered width at most if and only if is a subgraph of for some path .

Dujmović et al. [15] also showed it suffices to consider partitions of layered width 1.

[[15]] If a graph has an -partition of layered width for some graph of treewidth at most , then has an -partition of layered width 1 for some graph of treewidth at most . That is, if for some graph of treewidth at most and for some path , then for some graph of treewidth at most .

Dujmović et al. [15] proved the following result, which with Section 2.1, implies Section 1.

[[15]] Every planar graph has:

  1. an -partition of layered width for some planar graph of treewidth at most , and

  2. an -partition of layered width for some planar graph of treewidth at most .

Their proof is constructive and gives a simple quadratic-time algorithm for finding these partitions and corresponding layerings. The same authors proved the following generalisation of Sections 2.1 and 1 for graphs embeddable on other surfaces.444The Euler genus of the orientable surface with handles is . The Euler genus of the non-orientable surface with cross-caps is . The Euler genus of a graph is the minimum integer such that embeds in a surface of Euler genus . Of course, a graph is planar if and only if it has Euler genus 0; see [27] for more about graph embeddings in surfaces.

[[15]] Every graph of Euler genus is a subgraph of:

  1. for some graph of treewidth at most and for some path ;

  2. for some graph of treewidth at most and for some path ;

  3. for some graph of treewidth at most and some path .

Equivalently, every graph of Euler genus has:

  1. an -partition with layered width at most such that has treewidth at most ;

  2. an -partition with layered width at most such that has treewidth at most ;

  3. an -partition with layered width at most such that has treewidth at most .

Dujmović et al. [15] generalised Section 2.1 further as follows.555A graph is a minor of a graph if a graph isomorphic to can be obtained from a subgraph of by contracting edges. A class of graphs is minor-closed if for every graph , every minor of is in . A minor-closed class is proper if it is not the class of all graphs. For example, for fixed , the class of graphs with Euler genus at most is a proper minor-closed class. A graph is -apex if it contains a set of at most vertices such that is planar. A 1-apex graph is apex. A minor-closed class is apex-minor-free if some apex graph is not in .

[[15]] For every apex graph , there exists such that every -minor-free graph has an -partition with layered width 1 such that has treewidth at most . Equivalently, for some path .

3 New Results

Apex-minor-free graphs, addressed by Section 2.1, are the largest minor-closed class that admit bounded layered partitions [15]. However, the family of -planar graphs is not minor-closed. A graph obtained from a -planar graph by edge deletions and edge contractions may or may not be -planar. Indeed, Dujmović et al. [12] construct 1-planar graphs that contain arbitrarily large complete graph minors. Our results for -planar graphs are the first of this type for a non-minor-closed class.

The following result is the main theorem of the paper. Loosely speaking, it shows that if a graph admits bounded layered partitions, then so to does for every -shortcut system of .

[] Let be a graph having an -partition of layered width in which has treewidth at most and let be a -shortcut system for . Then has a -partition of layered width at most for some graph of treewidth at most .

Sections 2.1 and 3 immediately imply Section 1 in the introduction.

Using the relationship between -planar graphs and -shortcut systems along with a direct application of Section 3 we can obtain a slightly weaker version of Section 3, below. The minor modifications needed to obtain the stronger bound are described in Section 5.

Every -planar graph has an -partition of layered width at most in which has treewidth at most .

In the important special case of we obtain better constants and an additional property (planarity) of (see Section 5.1 for the proof):

Every 1-planar graph has an -partition of layered width at most 30 where is planar and has treewidth at most 3.

The definition of -planar graphs naturally generalises for other surfaces. A graph drawn on a surface is -plane if every edge of is involved in at most crossings. A graph is -planar if it is isomorphic to some -plane graph, for some surface with Euler genus at most . Grigoriev and Bodlaender [19] study approximation algorithms for -plane graphs. Section 1 immediately generalises as follows:

Every -planar graph is a subgraph of for some graph of Euler genus at most and some -shortcut system for .

We prove that -planar graphs admit bounded layered partitions.

Every -planar graph has an -partition of layered width at most in which has treewidth at most .

Again, a direct application of Section 3 using Section 2.1(b) implies Section 3 with a weaker bound on the layered width. We prove the stronger bound in Section 5.

Finally, we state the following corollary of Sections 2.1, 3 and 2.1. This is the most general result that follows from Section 3 and the work of Dujmović et al. [15].

For every apex graph and for all integers , there is an integer such that for every -minor-free graph and for every -shortcut system for , has an -partition of layered width 1 such that has treewidth at most ; that is for some path .

3.1 Previous Work on the Structure of -Planar Graphs

Prior to this work, the strongest structural description of -planar or -planar graphs was in terms of layered treewidth, which we now define. A layered tree-decomposition consists of a layering and a tree-decomposition of . The layered width of is . The layered treewidth of is the minimum layered width of any layered tree-decomposition of . Dujmović et al. [13] proved that planar graphs have layered treewidth at most 3, that graphs of Euler genus have layered treewidth at most , and more generally that a minor-closed class has bounded layered treewidth if and only if it excludes some apex graph. Dujmović et al. [12] show that every -planar graph has layered treewidth at most , and more generally that every -planar graph has layered treewidth at most .

If a graph class admits bounded layered partitions, then it also has bounded layered treewidth. In particular, Dujmović et al. [15] proved that if a graph has an -partition with layered width at most such that has treewidth at most , then has layered treewidth at most . What sets layered partitions apart from layered treewidth is that they lead to constant upper bounds on the queue-number and non-repetitive chromatic number, whereas for both these parameters, the best known upper bound obtainable via layered treewidth is ; see Section 6.

4 Shortcut Systems

The purpose of this section is to prove our main result, Section 3. This theorem shows how, given a -shortcut system of a graph , a -partition of can be used to obtain a -partition of where the layered width does not increase dramatically and the treewidth of is not much more than the treewidth of . Our main results for -planar graphs (Section 3) and -planar graphs (Section 3) follow; see Section 5.

For convenience, it will be helpful to assume that contains a length-1 -shortcut for every edge . Since is defined to be a supergraph of , this assumption has no effect on but eliminates special cases in some of our proofs.

For a tree rooted at some node , we we say that a node is a -ancestor of (and is a -descendant of ) if is a vertex of the path, in , from to . Note that each node is a -ancestor and -descendant of itself. We say that a -ancestor of is a strict -ancestor of if . The -depth of a node is the length of the path, in , from to . For each node , define

to be the maximal subtree of rooted at .

We begin with a fairly standard lemma about normalised tree decompositions.

nice-decomposition For every graph of treewidth , there is a rooted tree with and a -decomposition of with width that has following additional properties:

  1. For each node , the subtree is rooted at .

  2. For each edge , one of or is a -ancestor of the other.

Proof.

Begin with any width- tree decomposition of that uses some tree . Select any node , add a leaf , with , adjacent to and root at . Let be the function that maps each onto the root of the subtree . If is not one-to-one, then select some distinct pair with . Subdivide the edge between and its parent in by introducing a new node with . This modification reduces the number of distinct pairs with , so repeatedly performing this modification will eventually produce a tree-decomposition of in which is one-to-one.

Next, consider any node such that there is no vertex with . In this case, where is the parent of since any would have . In this case, contract the edge in , eliminating the node . Repeating this operation will eventually produce a width- tree-decomposition of where is a bijection between and . Renaming each node as gives a tree-decomposition with .

By the definition of , the tree-decomposition satisfies (T1). To see that satisifies (T2), observe that, if , then at least one of or is contained in for every node on the path from to in . If neither nor is an ancestor of the other, then some node on this path has -depth less than that of and . If this contradicts the fact that is the root of . If this contradicts the fact that is the root of . ∎

Figure 2: The sets , , and associated with and the ancestors of such that .

The following lemma shows how to interpret an -partition of and a tree-partition of as a hierarchical decomposition of ; refer to Figure 2.

Let be a graph; let be a layering of ; let be an -partition of of layered width at most with respect to where has treewidth at most ; and let be a tree-decomposition of satisfying the conditions of nice-decomposition. For each , let , , and . Then,

  1. is a partition of of layered width at most with respect to .

  2. For each , there is no edge with and .

  3. For each , there is a set of strict -ancestors of such that .

Before proving Section 4 we point out more properties that are immediately implied by it:

  1. for every .

  2. for every -ancestor of .

  3. for every -ancestor of .

Property (Y4) follows from the fact that is the union of several sets, one of which is . Property (Y5) follows from the definition of and the fact that . To show Property (Y6) first note that, by (Y5) it suffices to consider vertices . By definition, every vertex is adjacent, in , to a vertex . By (Y5), , so is either in or satisfies the condition , , and , so . In either case . Note that none of (Y4)–(Y6) depends on (Y3) (which is important, since (Y5) is used to establish (Y3) in the following proof).

Proof of Section 4.

Property (Y1) follows immediately from the fact that and the fact that has layered width at most with respect to . Property (Y2) is immediate from the definitions of and . In particular, is a separation of with .

To establish Property (Y3), consider some vertex . Since , there exists an edge with and . Since , for some -descendant of (possibly ). Since is a partition, for some . Since , we have . By (T2), one of or is a -ancestor of the other. Since and , (Y5) rules out the possibility that is a -ancestor of . Therefore, is a -ancestor of which is a -ancestor of . Let be the path in from to . For each , at least one of or is in . However, by (T1) is not contained in for any . Therefore for each . In particular, is contained in . Property (Y3) now follows from the fact that and contains . ∎

We are now ready to prove our main result, which we restate here for convenience:

See 3

Proof.

Apply Section 4 to and let , , , , , , , and be defined as in Section 4.

For a node , we say that a shortcut crosses if contains an internal vertex of , that is, and . We say that a vertex participates in if or contains a shortcut with and crosses . We let denote the set of nodes such that participates in .

For any there exists a (unique) node such that is a -ancestor of every node in .

Proof.

Let

Clearly is connected because is the union of (vertex sets of) paths in , each of which contains .

We claim that participates in a node if and only if . If participates in then either , so ; or for some shortcut that crosses , so for some . In the other direction, if , then either , so ; or where , and , so for a path that crosses .

Let . The connectivity of implies that is connected. Choose to be any member of that does not have a strict -ancestor in . Transitivity of the -ancestor relationship, (T2), and connectivity of implies that is a -ancestor of every node , as required. ∎

For each , we define the separator

Observe that is a partition of . We let denote the resulting quotient graph and we let in the obvious way, so that each is the vertex obtained by contracting in . (Nodes with do not contribute a vertex to .)

s-subset For every , .

Proof.

For the sake of contradiction, assume otherwise, so there exists some . By (Y4), , so . Therefore, contains a path , with , that crosses . The path contains a subpath such that and . Since and , (Y2), implies that for some . Now (Y3) implies for some strict -ancestor of . Therefore, either or crosses . But this implies that is a -ancestor of , which is a strict -ancestor of , contradicting the assumption that . ∎

Next we show that has small layered width with respect to :

general-width For each and each , .

Proof.

We count the number of vertices in by upper-bounding the number of vertices contributed to by each vertex . Refer to Figure 3. If and no path in includes as an internal vertex then contributes only one vertex, itself, to .

Figure 3: A path containing an internal vertex .

Otherwise, consider some path that contains as an internal vertex. If , then contributes at most vertices to . If , then contributes at most vertices to . If for , then contributes at most vertices to .

For any , the number of vertices is at most . Each such vertex is an internal vertex of at most paths in . Therefore,

i-ancestor For each edge , one of or is a -ancestor of the other.

Proof.

Suppose, for the sake of contradiction, that neither nor is a -ancestor of the other. Since , contains an edge with and . Since , contains a -shortcut . By s-subset, and . By (Y5), if neither nor is a -ancestor of the other, then and are disjoint. By (Y3) and are also disjoint. By (Y2) contains an internal vertex . By (Y3), for some strict -ancestor of . But this implies that so for some -ancestor of , contradicting the assumption that . ∎

general-bag-size The graph has a tree-decomposition in which every bag has size at most .

Proof.

For the tree-decomposition of we use the same tree used in the tree-decomposition of . For each node of , contains as well as every -ancestor of such that contains an edge where is a -ancestor of (including the possibility that ). i-ancestor ensures that, for every edge , and the connectivity of is obvious. Therefore is indeed a tree-decomposition of . It remains to bound the size of each bag .

Consider an arbitrary node where is the path from the root of to . To avoid triple-subscripts in what follows, we abuse notation slightly by using , , and , as shorthands for , and , respectively.

If , it is because for some -descendant of . This implies contains an edge with and . This implies that contains a -shortcut . Let be the second-last vertex of (so ).

Since , at least one of the following is true:

  1. contains a -shortcut that has an internal vertex in ; or

  2. . In this case, we define to be the path of length 0 that contains only .

Let denote the first vertex of contained in .

Let be the path that begins and then follows the subpath of that begins at and ends at . For each , let , and let . Note that is a non-increasing sequence and is a sequence of nodes of whose distance from the root, , of is non-increasing.

By definition, . We claim that , i.e., . To see this, first observe that, for each , since, otherwise, an internal vertex of belong to , which would imply (by (Y3)) that for some , contradicting the assumption that . Therefore . To see that , observe that either or contains an internal vertex in . By (Y1) and the defnition of , does not contain , so .

Let denote the supergraph of with vertex set and in which if and only there exists some such that . We claim that is a lazy walk666A lazy walk in a graph is a walk in the pseudograph obtained by adding a self loop to each vertex of . in . Indeed, if for some then this is precisely because but . By definition, for some -descendant of . By (Y2), so by (Y3) for some strict -ancestor of . Since , . By (T1), and . Since is on the path from to in this implies that . Therefore as claimed.

Thus, is a lazy walk in of length where the distance between and the root of is non-decreasing. By removing repeated vertices this gives a path in the directed graph obtained by directing each edge from its -descendant towards its -ancestor . Finally, we are in a position to appeal to [30, Lemma 24] which states that the number of nodes in that can be reached from any node by a directed path of length at most is at most . ∎

At this point, the proof of Section 3 is almost immediate from general-width and general-bag-size, except that the layering of may not be a valid layering of . In particular, may contain edges with and for any . To resolve this, we use a new layering in which . This increases the layered width given by general-width from to . Therefore has an -partition of layered width at most