Planar Graphs have Bounded Queue-Number

Abstract

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number.Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. (2004) that graphs in a proper minor-closed class have low treewidth colourings.

Improved bounds for centered colorings

“…Low treedepth colorings are a central tool for designing parametrized algorithms in classes of bounded expansion. For example, Pilipczuk and Siebertz [14] showed that when C is a class of graphs avoiding a fixed graph as a minor, then given graphs H and G on p and n vertices, respectively, where G is in C, it can be decided whether H is a subgraph of G in time 2 O(p log p) · n O(1) and space n O (1) . The algorithm witnessing this statement starts with a computation of a p-centered coloring of G with p O(1) colors, and for each p-tuple of colors it applies a procedure to solve this problem for graphs G of treedepth at most p. The results of our paper imply a corresponding algorithm for the case where C is a class of graphs avoiding a fixed graph as a topological minor.…”

“…The previously best known bound was O(p 19 ) which was given by Pilipczuk and Siebertz [14]. A key tool responsible for the improvement of the exponent is a brand new structure theorem for planar graphs due to Dujmović et al [1] which has its roots in [14]. In Section 2, we give a precise statement of the theorem and we show how to use it to color a planar graph in a p-centered way with O(p) · f (p) colors where f (p) is the maximum number of colors we need in a p-centered coloring of planar graphs of treewidth at most 3.…”

An Experimental Study of a 1-Planarity Testing and Embedding Algorithm

“…A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. An n-vertex 1-planar graph has O(n) edges [43], O( √ n) separators and hence O( √ n) treewidth [25,30], and O(1) stack and queue number [4,8,26]. Despite these (and other) similarities with planar graphs, recognizing whether a graph is 1-planar is an NP-complete problem, in contrast with the well-known efficient algorithms for testing planarity.…”

Mixed Linear Layouts: Complexity, Heuristics, and Experiments

“…Klawitter et al [10] surveyed the literature and performed an experimental study on several stateof-the-art book drawing algorithms aiming to minimize the number of crossings in layouts on a fixed number of stack pages. Conversely, for queue layouts it was a longstanding open question whether planar graphs have bounded queue number [8]; this was recently answered positively by Dujmović et al [4].…”