Weak Coloring Numbers of Intersection Graphs

03/31/2021 ∙ by Zdeněk Dvořák, et al. ∙ KIT Charles University in Prague 0

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number k, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in k steps only few vertices with lower index in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in the structural and algorithmic graph theory. Recently, the first author together with McCarty and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in ℝ^d, such as homothets of a centrally symmetric compact convex object, or comparable axis-aligned boxes. In this paper, we prove upper and lower bounds for the k-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in k, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).

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1 Introduction

All the graphs we consider are finite, simple and undirected. For the basic graph theoretic notions used in this paper, see [1].

Given a linear ordering of vertices of a graph and an integer , a vertex is weakly -reachable from a vertex if and there exists a path in from to of length at most with all internal vertices greater than , and strongly -reachable if there exists such a path with all internal vertices greater than . Let and denote the sets of vertices that are weakly and strongly -reachable from , respectively. We define weak and strong coloring numbers for a given ordering as

The weak and strong coloring numbers of a graph are then obtained by minimizing over all linear orderings of .

Note that for , both and consist of the neighbors of that precede it in the ordering , and thus coincide with the coloring number of the graph , equal to the degeneracy of plus one.

One can easily check the following observations:

for any .

Let be a graph. Let and be the chromatic number and the maximum degree of . One can easily check the following relation between those graph parameters and :

Weak coloring numbers were introduced by Kierstead and Yang [7] and were used in particular to study marking and coloring games on graphs. They also showed the following inequality:

for any and a graph .

Zhu [13] showed that the weak and strong coloring numbers can be also used to study important notions of sparsity in a graph and characterize classes of graphs with bounded expansion and nowhere dense classes. A graph class has bounded expansion if there is a function such that for every and for any system of pairwise vertex-disjoint subgraphs of of radius at most , the minor obtained by contacting each into a vertex and deleting all other vertices of has average degree at most . Classes of bounded expansion include planar graphs and more generally all proper classes closed under taking minors or topological minors. See [9, 10] for more information on this topic.

Before proceeding to our results we mention some known bounds on maximum of , , for some specific classes of graphs . For the class of outerplanar graphs the bound is [6]; for the class of planar graphs the upper bound is [11] and the lower bound is [6]; for the class of graphs of Euler genus the upper bound is [11]; for the class of graphs of tree width at most the upper bound is [5].

2 Our results

Let be a finite set of objects in . The intersection graph of is the graph with and with if and only if . For an integer , we say that the set is -thin if every point of is contained in the interior of at most objects from ; in case , we say is a touching representation of . For example, a famous result of Koebe [8] states that a graph is planar if and only if it has a touching representation by balls in . As observed in [3], there is a very natural way of bounding the strong coloring numbers for thin intersection graphs of certain classes of objects by ordering the vertices in a non-increasing order according to the size of the objects that represent the vertices. In particular, this approach works in case the objects in are

  • scaled and translated copies of the same centrally symmetric compact convex object (this includes intersection graphs of balls and of axis-aligned hypercubes); or

  • -ball-like for some real number , i.e., every is a compact convex set satisfying , where is the ball in of radius and is the maximum distance between any two points of ; or

  • comparable axis-aligned boxes, i.e, is a set of axis-aligned boxes with the additional property that for every , a translation of is a subset of or vice versa.

As we are going to build on this argument, let us give a sketch of it. A linear ordering of a finite set of compact objects is sizewise if for all such that , we have .

Lemma 1.

Let and be positive integers. Let be a -thin finite set of compact convex objects in and let be the intersection graph of . Let be a sizewise linear ordering of . For each integer ,

  • if consists of scaled and translated copies of the same centrally symmetric object, or if is a set of comparable axis-aligned boxes, then , and

  • if consists of -ball-like objects for a real number , then .

Proof.

Consider a vertex ; we need to provide an upper bound on . For any , in case (a) let be the object obtained by scaling by the factor of , with the center of being the fixed point; i.e., . In case (b), let be a ball of radius centered at an arbitrarily chosen point of .

For each , observe that , as is joined to through a path with at most internal vertices, each represented by an object smaller or equal to in size. In case (a), observe that there exists a translation of such that and . In case (b), let be a scaled translation of such that , , and . Note that in the former case we have , and in the latter case we have

In either case, observe that , and since is -thin, we have

Therefore, in case (a) and in case (b). ∎

That is, the strong coloring numbers of these graph classes are polynomial in , with a uniform ordering of vertices that works for all values of . For weak coloring numbers, a general upper bound is as follows.

Observation 2.

For any graph , a linear ordering of its vertices, and an integer ,

In particular, if there exists such that for every , then for every .

For graphs from the classes described in Lemma 1, we obtain an exponential bound on the weak coloring numbers, more precisely in case (a) and in case (b).

Joret and Wood (see [4]) conjectured that every class of graphs with polynomial strong coloring numbers also has polynomial weak coloring numbers. This turns out not to be the case; Grohe et al. [5] showed that the class of graphs obtained by subdividing each edge of the graph the number of times equal to its treewidth has superpolynomial coloring numbers. However, one could still expect this conjecture to hold for “natural” graph classes, and thus we ask whether the weak coloring numbers are polynomial for the graph classes described in Lemma 1. On the positive side, we obtain the following result.

Theorem 3.

Let and be positive integers. Let be a -thin finite set of compact convex objects in and let be the intersection graph of . Let be a sizewise linear ordering of . For each integer :

  • If consists of scaled and translated copies of the same centrally symmetric object, then

  • If consists of -ball-like objects for a real number , then

Moreover, there exists (depending only on ) such that if consists of balls, then for every ,

This theorem is qualitatively tight in several surprising aspects, summarized in the following result.

Theorem 4.

For every positive integer :

  • There exists a touching graph of comparable axis-aligned boxes in such that .

  • For every , there exists a -thin set of axis-aligned squares in whose intersection graph satisfies .

  • For every , the graph can also be represented as a touching graph of axis-aligned hypercubes in .

That is:

  • The (rather natural) class of touching graphs of comparable axis-aligned boxes in has polynomial strong coloring numbers but exponential weak coloring ones. Let us remark that touching graphs of rectangles in are obtained from planar graphs by adding crossing edges into faces of size four (when four of the boxes share corners), and such graphs have polynomial weak coloring numbers (this follows e.g. from their product structure [2]).

  • Unlike in Lemma 1, in dimension at least two the dependence of the weak coloring numbers in Theorem 3 on the thinness must indeed be in the exponent, and not just in the multiplicative constant. Let us also remark that -thin intersection graphs of intervals in are interval graphs of clique number at most . As was pointed to us by Gwenaël Joret, any interval graph of clique number satisfies , as shown by an ordering obtained by placing first the vertices of a maximal system of pairwise disjoint cliques of size and then recursively processing the remainder of the graph which has clique number smaller than .

  • In the case (a) of Theorem 3, and in particular for the touching graphs of axis-aligned hypercubes, the exponent must be exponential in the dimension, in a contrast to the case of touching graphs of balls.

3 Upper bounds

In order to prove Theorem 3 for all the classes at once, let us formulate an abstract graph property on which the proof is based. For a graph , a function and , let us define as the minimum of over all paths from to in . For a function and positive integers and , we say that has the property if

  • for each and integers and , there are at most vertices such that and , and

  • for each and each positive integer , every sequence , , … of distinct vertices of such that and for each has length at most .

Let us remark that implies for every , and (i) implies (ii) with and . The following lemma is proved similarly to Lemma 1.

Lemma 5.

Let and be positive integers. Let be a -thin finite set of compact convex objects in and let be the intersection graph of . For , let .

  • If consists of scaled and translated copies of the same centrally symmetric object, then has the property .

  • If consists of -ball-like objects for , then has the property .

  • If consists of balls, then there exists such that has the property .

Proof.

Consider a vertex and integers and . For any , in cases (a) and (c) let be the object obtained by scaling by the factor of , with the center of being the fixed point. In case (b), let be a ball of radius centered at an arbitrarily chosen point of . Let be the set of vertices such that and . Observe that for any , we have . Let be a scaled translation of such that , , and . For each , in cases (a) and (c), we have

and in case (b) we have

In either case, we have , and since is -thin, it follows that

in cases (a) and (c), and

in case (b). Hence, the part (i) of the property is verified, and by the observations made before the lemma, this finishes the proof for the cases (a) and (b).

Let us now consider the part (ii) in case (c). Let

be a half-space whose boundary hyperplane touches

and is otherwise disjoint from . There exists such that ; let us fix smallest such . For , let be a ball touching of radius . I.e. . Note that

and thus there exists such that ; let us fix smallest such .

Consider a sequence , , …, of distinct vertices of such that and for each . In particular, note that for each . From the observation made in the first paragraph of the proof, we have , and it follows that

Since is -thin and is an integer, this implies , verifying the part (ii) of the property . ∎

To bound the weak coloring numbers, we need the following result about graphs of bounded pathwidth which appears in a stronger form (for treewidth) in van den Heuvel et al. [12]. For us, it is convenient to state the result as follows (without explicitly defining pathwidth), and thus we include the proof for completeness. A path in a graph with a linear ordering of vertices is decreasing if . For each , we define as the set of vertices reachable from by decreasing paths of length at most .

Lemma 6.

Let and be non-negative integers. Let be a linear ordering of the vertices of a graph . If for every , at most vertices have a neighbor , then for every .

Proof.

Without loss of generality, we assume that if and , then is also adjacent to all vertices such that . Indeed, adding such an edge does not violate the assumptions and can only increase .

The proof is by induction on . Note that , and thus we can assume . If no neighbor of is smaller than , then , and thus the claim of the lemma holds. Hence, we can assume has such a neighbor, and in particular . Let be the smallest neighbor of . Let be the subgraph of induced by the vertices greater than and smaller or equal to . Since is adjacent to all the vertices of , note that for each , at most vertices of have a neighbor in .

Consider now a vertex , and let be a decreasing path of length at most from to . If , then is also a decreasing path in , and thus . Note that by the induction hypothesis. If , consider the edge of such that and . Note that is not adjacent to by the minimality of , and thus . Moreover, by the assumption made in the first paragraph, . Hence, is reachable from by the decreasing path of length at most starting with and continuing along , and thus . If , then we also have . By the induction hypothesis, we have .

Therefore,

We use the following corollary, obtained by applying Lemma 6 to the graph obtained by contracting each interval to a single vertex.

Corollary 7.

Let , , and be non-negative integers. Let be a linear ordering of vertices of a graph , and let be a partition of into consecutive intervals in this ordering, where for every , , and , we have (note the reverse ordering of the indices). Suppose that for each , we have and there are at most indices such that a vertex of has a neighbor in . Then for each .

Theorem 3 now follows from Lemma 5 and the following theorem.

Theorem 8.

Let be a function and let and be positive integers. For a graph and a function , let be a linear ordering of such that if , then . If has the property , then

for every integer .

Proof.

Consider any integer and a vertex ; we are going to bound the number of vertices weakly -reachable from . Note that for , consists of the vertices such that and , and thus by the part (i) of the property with and . Hence, we can assume that .

Let be the graph with the vertex set , such that for with , we have if and only if there exists a path of length at most in from to such that and all the internal vertices of the subpath of between and are greater than . Let denote the minimum length of the subpath between and over all paths satisfying these conditions. Observe that for every edge of , there exists a decreasing path from containing the edge such that . Moreover, .

For , let consist of the vertices such that ; in particular, . Let and further partition into , where consists of the vertices with for . Consider any vertex . Since is weakly -reachable from and , we have . Moreover, , and thus by the part (i) of the property with and , we conclude for each . Hence, we have .

Let be all indices such that and for each , a vertex has a neighbor for each . For , since there exists a decreasing path from containing the edge such that , there exists a path in from to of length at most such that for every internal vertex of . Consequently, we have for . Moreover, note that , and thus . By part (ii) of the property , we conclude that .

Hence, Corollary 7 implies that

for each . ∎

4 Lower bounds

Figure 1: The graph and its intersection representation by squares.

It is relatively easy to construct intersection graphs with large weak coloring numbers with respect to a fixed coloring. The following construction (illustrated in the top part of Figure 1) enables us to turn such graphs into graphs that have large weak coloring numbers with respect to every ordering. Let be a graph and a linear ordering of its vertices. Let be the vertices of . Let be a positive integer and let be the complete rooted -ary tree of depth . For , let be the set of vertices of at distance exactly from the root. The graph has vertex set , with vertices and adjacent if and only if , , and is an ancestor of in or vice versa. We say that is the scaffolding of .

Lemma 9.

Let and be positive integers. Let be a graph and a linear ordering of its vertices. Suppose that for each , the graph is connected and has diameter at most . Then

Proof.

Consider any linear ordering of the vertices of . Let be the scaffolding of and suppose first that there exists a non-leaf vertex such that all children , …, of in are smaller than in the ordering . For , let be the subgraph of induced by , , and all descendants of in . Let be the vertex of such that ; since the graph has diameter at most , every vertex of is at distance at most from . Since , we conclude that a vertex of distinct from is weakly -reachable from . Since this is the case for each and the subgraphs , …, intersect only in , it follows that

Hence, we can assume that each non-leaf vertex of has a child which is greater than in the ordering . Consequently, contains a path from the root to a leaf such that . The subgraph of induced by with ordering is isomorphic to with ordering , and thus

Moreover, assuming has a sufficiently generic representation by comparable axis-aligned boxes, we can also find such a representation for . Given an axis-aligned box in and , let denote the length of in the -th coordinate. We say that a sequence , …, of axis-aligned boxes is -shrinking if holds for . See the bottom part of Figure 1 for an illustration of the following construction.

Lemma 10.

Let , and be positive integers. Let be a -thin finite set of comparable axis-aligned boxes in and let be the intersection graph of . Let be the scaffolding of . Let be a sizewise linear ordering of and let , …, be the sequence of vertices of in this order. If this sequence is -shrinking, then is the intersection graph of a -thin set of comparable axis-aligned boxes in , where for and , is the product of with an interval of length .

Proof.

Let be small enough so that holds for . For each non-leaf vertex of , assign labels , …, to the edges from to the children of in any order; let denote the label assigned to the edge . For a vertex of , if is the path in from the root to , then let . Note that is an ancestor of a vertex in if and only if is a prefix of . Let , and let be the interval . Observe that if is an ancestor of a vertex in , then , and if is neither an ancestor nor a descendant of in , then .

Hence, letting each vertex at distance from the root of be represented by the box in , we obtain a -thin intersection representation of as described in the statement of the lemma. ∎

To verify the assumptions of Lemma 9, the following concept is useful. Let be a linear ordering of vertices of a graph . A decreasing spanning tree is a spanning tree of rooted in the maximum vertex such that any path in starting in the root is decreasing.

Lemma 11.

Let be an integer. Let be a linear ordering of vertices of a graph . If has a decreasing spanning tree of depth at most , then , and for each , the graph is connected and has diameter at most .

Proof.

Let be the maximum vertex of . Since is decreasing and has depth at most , we have . Moreover, for each , letting , observe that for each , all ancestors of also belong to . Hence, is a spanning tree of of depth at most , and thus is connected and has diameter at most . ∎

We now find some basic graphs to which we can apply the construction.

Figure 2: The construction from Lemma 12.
Lemma 12.

For all integers and , there exists a graph with vertices represented as the touching graph of an -shrinking sequence of comparable axis-aligned rectangles in , such that has a spanning tree of depth at most decreasing in the sizewise ordering.

Proof.

We proceed by the induction on . For each , we construct a representation of where the last vertex is represented by a unit square and the rest of the representation is contained in the lower left quadrant starting from the middle of the upper side of . The second coordinate (relevant for the definition of an -shrinking sequence) is the horizontal one. In the vertical coordinate, all rectangles have length . See Figure 2 for an illustration of the construction.

The graph is a single vertex represented by . For , to obtain a representation of , we scale the representation of in the horizontal direction by the factor of and place it so that its upper right corner is the middle of the lower side of . Then we add another copy of a representation of , scaled in the horizontal direction so that all its rectangles are more than times longer than the already placed ones and so that when we place its upper right corner at the upper left corner of , their interiors are disjoint from the already placed rectangles.

Observe that contains a spanning complete binary tree of depth rooted in , with the vertices along each path from the root increasing in size, and thus decreasing in the sizewise ordering. ∎

Figure 3: The construction from Lemma 13.
Lemma 13.

For all integers and , there exists a graph with