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Separating Layered Treewidth and Row Treewidth

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Separating Layered Treewidth and Row Treewidth Separating Layered Treewidth and Row Treewidth Prosenjit Bose1 Vida Dujmovic´2 Mehrnoosh Javarsineh1 Pat Morin1 David R. Wood3 May 5, 2021 Abstract. Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It fol- lows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In par- ticular, for every integer k we describe a graph with layered treewidth 1 and row treewidth k. We also prove an analogous result for layered pathwidth and row pathwidth. arXiv:2105.01230v1 [math.CO] 4 May 2021 1School of Computer Science, Carleton University, Ottawa, Canada (jit,[email protected]). Research supported by NSERC. 2School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada ([email protected]). Research supported by NSERC and the Ontario Ministry of Research and Innovation. 3School of Mathematics, Monash University, Melbourne, Australia ([email protected]). Research supported by the Australian Research Council. 1 Introduction Treewidth is a graph parameter that measures how similar a given graph is to a tree; it is of fundamental importance in structural and algorithmic graph theory; see the surveys [3, 23, 29]. Layered treewidth is a variant of treewidth introduced independently by Dujmovic,´ Morin, and Wood [15] and Shahrokhi [34]; see Section2 for the definition. A key property is that layered treewidth is bounded on planar graphs but treewidth is not. In particular, planar graphs have layered treewidth at most 3 [15], but the n n grid graph has treewidth × n. Layered treewidth has been used in upper bounds on several graph parameters includ- ing queue-number [13, 15], stack number [13], boxicity [33], clustered chromatic num- ber [27], generalised colouring numbers [35], asymptotic dimension [4], as well for results in intersection graph theory [34]. Row treewidth1 is a refinement of layered treewidth, implicitly introduced by Duj- movic,´ Joret, Micek, Morin, Ueckerdt, and Wood [14], who proved that planar graphs have row treewidth at most 8. This result and its generalisations have been the key to solv- ing several open problems, regarding queue-number [14], nonrepetitive chromatic num- ber [12], universal graphs [5, 11, 21], centred colouring [7], graph drawing [14, 28], and vertex ranking [6]. Layered and row treewidth are closely related in that the layered treewidth of a graph is at most its row treewidth plus 1, and a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, a fundamental open problem is whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. Theorem 1. For each k N there is a graph with layered treewidth 1 and row treewidth k. 2 This result is proved in Section3. In Section4 we use a key lemma from the proof of Theorem1 to prove a result about the relationship between layered treewidth and queue- number. Layered pathwidth is a graph parameter analogous to layered treewidth, first studied by Bannister, Devanny, Dujmovic, Eppstein, and Wood [1] and Dujmovic,´ Eppstein, Joret, Morin, and Wood [10]. Row pathwidth is defined in a similar way to row treewidth. We prove the following analogue of Theorem1 for layered pathwidth and row pathwidth. Theorem 2. For each k N there is a tree with layered pathwidth 1 and row pathwidth k. 2 2 Definitions We consider finite, undirected, simple graphs and use standard graph theory terminol- ogy [8]. 1The name “row treewidth” is an original contribution of the present paper. 1 Minors A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. A graph class is minor-closed if for every graph G G , every minor of G is in . A graph class is proper if some graph is not in .A K 2 G G G G t model in a graph G is a set X ;:::;X of pairwise-disjoint connected subgraphs in G, such f 1 tg that there is an edge of G between X and X for all distinct i;j 1;:::;t . Clearly, K is a i j 2 f g t minor of G if and only if G contains a Kt model. Treewidth and Pathwidth A tree decomposition of a graph G is a collection (B : x V (T )) of subsets of V (G) called T x 2 bags indexed by the nodes of a tree T such that (i) for each v V (G), the induced subgraph 2 T [ x V (T ): v B ] is nonempty and connected; and (ii) for each edge vw E(G), there f 2 2 xg 2 exists x V (T ) such that v;w B . The width of a tree decomposition is the size of 2 f g ⊆ x its largest bag, minus 1. The treewidth tw(G) of a graph G is the minimum width of any tree decomposition of G. If = (B : x V (P )) is a tree decomposition of G and P is P x 2 a path, then is a path decomposition of G. The pathwidth pw(G) of a graph G is the P minimum width of any path decomposition of G. For each k N, the graphs with treewidth 2 at most k form a minor-closed class, as do the graphs with pathwidth at most k. Note that tw(K ) = pw(K ) = n 1. n n − Layered Treewidth and Pathwidth A layering of a graph G is a partition of V (G) into a sequence of sets (L ;:::;L ) such that L 0 n for any edge vw E(G), if v Li and w Lj then i j 6 1. For example, if r is a vertex in a 2 2 2 j − j connected graph G, and Li is the set of vertices at distance i from r in G, then (L0;L1;:::) is the breadth-first layering of G rooted at r.A layered tree decomposition of a graph G consists of a pair ( ; ) where = (L ;:::;L ) is a layering of G and = (B : x V (T )) is a tree L T L 1 n T x 2 decomposition of G. The (layered) width of ( ; ) is the size of largest intersection between L T a layer and a bag; that is, max B L : x V (T );i 1;:::;n . The layered treewidth ltw(G) fj x \ ij 2 2 f gg of G is the minimum width of any layered tree decomposition of G. This definition was introduced independently by Shahrokhi [34] and Dujmovic´ et al. [15]. The latter authors proved that ltw(G) 6 3 for every planar graph G; more generally that ltw(G) 6 2g + 3 for every graph G of Euler genus g; and that a minor-closed class has G bounded layered treewidth if and only if some apex graph2 is not in . For an arbitrary G proper minor-closed class , Dujmovic´ et al. [15] showed that every graph in has a tree- G G decomposition in which each bag has a bounded set of vertices whose deletion leaves a sub- graph with bounded layered treewidth. This version of Robertson and Seymour’s Graph Minor Structure Theorem has proved to be very useful [4, 26, 27]. If ( ; ) is a layered tree decomposition of G and is a path decomposition, then ( ; ) L P P L P is a layered path decomposition of G. The layered pathwidth of G is the minimum width of any layered path decomposition of G. This parameter was introduced by Bannister et al. [1], 2A graph G is apex if G v is planar for some vertex v. A graph G is an apex-forest if G v is a forest for − − some vertex v. 2 who proved that every outerplanar graph has layered pathwidth at most 2 (amongst other examples). Row Treewidth and Pathwidth The cartesian product of graphs A and B, denoted by A B, is the graph with vertex set V (A) V (B), where distinct vertices (v;x);(w;y) V (A) V (B) are adjacent if: v = w and × 2 × xy E(B); or x = y and vw E(A). The direct product of A and B, denoted by A B, is the 2 2 × graph with vertex set V (A) V (B), where distinct vertices (v;x);(w;y) V (A) V (B) are × 2 × adjacent if vw E(A) and xy E(B). The strong product of A and B, denoted by AB, is the 2 2 union of A B and A B. × The row treewidth rtw(G) of a graph G is the minimum treewidth of a graph H such that G is isomorphic to a subgraph of H P , where P is the 1-way infinite path. This 1 1 definition is implicit in the work of Dujmovic´ et al. [14] who proved that rtw(G) 6 8 for every planar graph G; more generally, that rtw(G) 6 2g + 9 for every graph G of Euler genus g; and that a minor-closed class has bounded row treewidth if and only if some G apex graph is not in . For an arbitrary minor-closed class , Dujmovic´ et al. [14] showed G G that every graph in has a tree-decomposition in which each bag has a bounded set of G vertices whose deletion leaves a subgraph with bounded row treewidth.
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