Layered Separators for Queue Layouts, 3D Graph Drawing and Nonrepetitive Coloring

Abstract

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 families. 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 breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. 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 to all proper minor-closed families, with a O(log n) bound on the nonrepetitive chromatic number, and a log O(1) n bound on the queue-number. Only for planar graphs were log O(1) n bounds previously known. Our results imply that every graph from a proper minor-closed class has a 3-dimensional grid drawing with n log O(1) n volume, whereas the previous best bound was O(n 3/2 ). Readers interested in the full details should consult arXiv:1302.0304 and arXiv:1306.1595, rather than the current extended abstract.

Stack and Queue Layouts via Layered Separators

“…Together with the fact that planar graphs have layered 2-separators [13,19], these results imply an O(log n) bound for the queue-number of planar graphs, improving on a earlier result by Di Battista et al [10]. The polylog bound on the queue-number of planar graphs extends to all proper minor-closed families of graphs [15,16]. Our approach to prove Theorem 1 also gives a new proof of the following result (without using track layouts).…”

“…Layered separators are a key tool for proving our results. They have already led to progress on long-standing open problems related to 3D graph drawings [11,15] and nonrepetitive graph colourings [13]. A layering {V 0 , .…”

“…More precisely, Dujmović et al[12] proved that (g, k)-planar graphs have layered treewidth at most (4g + 6)(k + 1) and (g, d)-map graphs have layered treewidth at most (2g + 3)(2d + 1). Just as the graphs of treewidth t have (classical) separators of size t − 1, so do the graphs of layered treewidth have layered -separators[15,16].…”

Track Layouts, Layered Path Decompositions, and Leveled Planarity

“…A layering of a graph is a partition of the vertices into a sequence of disjoint subsets (called layers) such that each edge joins vertices in the same layer or consecutive layers. One way, but not the only way, to obtain a layering is the breadth first layering in which we partition the vertices by their distances from a fixed starting vertex [17,18]. We emphasis that a layering does not specify an ordering of the vertices within each layer, so there is no notion of edge crossings in a layering.…”

“…The layered width of layered tree decomposition is the size of the largest intersection of a bag with a layer. The layered treewidth of a graph G is the minimum layered width of a tree-decomposition of G. Dujmović, Morin, and Wood [17,18] introduced layered treewidth and proved that every planar graph has layered treewidth at most 3, that every graph with Euler genus g has layered treewidth at most 2g + 3, and more generally that a minor-closed class has bounded layered treewidth if and only if it excludes some apex graph. Dujmović, Eppstein, and Wood [11,12] showed that layered treewidth is of interest well beyond minor-closed classes.…”