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 [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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2013
- DOI:
- 10.48550/arXiv.1302.0304
- arXiv:
- arXiv:1302.0304
- Bibcode:
- 2013arXiv1302.0304D
- Keywords:
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- Computer Science - Computational Geometry;
- Computer Science - Data Structures and Algorithms;
- 05C