Graph Layouts via Layered Separators

TL;DR

This paper proves that all planar graphs have a track number and queue number at most logarithmic in the number of vertices, improving previous bounds and enabling efficient 3D grid drawings.

Contribution

It establishes the first logarithmic upper bounds on the track and queue numbers of planar graphs using novel separator techniques.

Findings

Planar graphs have queue and track numbers at most O(log n).

Planar graphs can be embedded in 3D grid with O(n log n) volume.

Improves previous polynomial bounds on these parameters.

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 [Foundations of Computer Science, (FOCS '10), pp. 365--374] who proved the first sub-polynomial bounds on the queue number and track number of planar graphs. Specifically, they obtained O(log^2 n) queue number and O(log^8 n) track number bounds for planar graphs. The result also implies that every planar graph has a 3D crossing-free grid drawing in O(n log n) volume. The proof uses a non-standard type of graph separators.