Stack and Queue Layouts via Layered Separators

Abstract

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed g and k, we show that every n-vertex graph that can be embedded on a surface of genus g with at most k crossings per edge has stack-number O(log n); this includes k-planar graphs. The previously best known bound for the stack-number of these families was O( √ n), except in the case of 1-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that n-vertex graphs that admit constant layered separators have O(log n) stack-number.

A Survey on Graph Drawing Beyond Planarity

“…It is worth remarking that several papers present interesting bounds of graph parameters for 1-planar and k-planar graphs, which shed some light on the structure of these graphs and thus which may be of interest for the design of fixed-parameter tractable recognition algorithms. For example, kplanar graphs on n vertices have O(log n) book thickness [115], O( √ kn) treewidth and O(k) layered treewidth [114], and bounded expansion [167] (see [159] for more results).…”

“…If no two edges cross, a k-page drawing is called k-page book embedding (or k-stack layout). The minimum value of k such that a graph G has a k-page book embedding is the book thickness (or stack number ) of G. An O(log n) upper bound on the book thickness of k-planar graphs has been recently proved [115], while for 1-planar graphs O(1) upper bounds are knwon [11,38].…”

Planar Graphs of Bounded Degree Have Bounded Queue Number

“…If the input is a k-planar graph of bounded degree, the obtained planarization is also of bounded degree, and we obtain the following corollary. Note that, Dujmović and Frati [11] proved that k-planar graphs have queue number O(log n).…”

Improper colourings inspired by Hadwiger's conjecture

“…The layered treewidth of a graph G is the minimum layered width of a tree decomposition of G. Layered treewidth was introduced independently by Dujmović et al [18] and Shahrokhi [54]. Applications of layered treewidth include nonrepetitive graph colouring [18], queue and track layouts [18], graph drawing [4,18], book embeddings [17], and intersection graph theory [54].…”