Graph layouts via layered separators

Abstract

A k-queue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queue-number (track-number) of a graph G, is the minimum k such that G has a k-queue (k-track) layout.This paper proves that every n-vertex planar graph has track number and queue number at most O(log⁡n). This improves the result of Di Battista, Frati, and Pach (2013) [5] who proved the first sub-polynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log2⁡n) queue number and O(log8⁡n) track number bounds for planar graphs.The result also implies that every planar graph has a 3D crossing-free grid drawing in O(nlog⁡n) volume. The proof uses a non-standard type of graph separators.