Layered separators in minor-closed graph classes with applications

Abstract

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor.We use layered separators to prove O(log n) bounds for a number of problems where O( √ n) was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a O(log n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a log O(1) n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were log O(1) n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with n log O(1) n volume, whereas the previous best bound was O(n 3/2 ).

Untitled

“…A graph with a k-queue layout is called a k-queue graph. The queue-number of a graph G is the minimum integer k such that there is a k-queue layout of G. See [13,29,30,16,15,8,14,17] and the references therein for results on queue layouts. A d-monotone bipartite graph has queue-number at most d, since using the above notation, edges in a monotone matching do not cross in the vertex ordering (v 1 , .…”

“…That is, there are no edges vw and xy in G with v ≺ x in some track V i , and y ≺ w in some track V j . The track-number is the minimum integer k for which there is a k-track layout of G. See [13,15,14,9,17] and the references therein for results on track layouts. We prove the following: Theorem 1.5.…”

“…The best upper bound on the queue-number of planar graphs is O(log n) due to Dujmović [11]; see [14] for recent extensions. The results in this paper may be good news for this problem, in the sense that planar graphs are a well-behaved class (containing no expanders), whereas graphs with bounded queue-number are a rich class of graphs (containing expanders).…”

Genus, Treewidth, and Local Crossing Number

“…We prove our upper bounds by using the concept of layered treewidth [1], and we prove matching lower bounds by finding (g, k)-planar graphs without large separators and using the known relations between separator size and treewidth.…”

“…1 For instance, Guy et al [7] investigated the local crossing number of toroidal embeddings-in this notation, the (2, k)-planar graphs. We again determine an optimal bound on the treewidth of such graphs.…”