The geometric thickness of low degree graphs

Authors:
Christian A. Duncan;David Eppstein;Stephen G. Kobourov
Affiliations:
University of Miami;University of California, Irvine, CA;University of Arizona
Venue:
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Year:
2004

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Abstract

We prove that the geometric thickness of graphs whose maximum degree is no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph. In our proofs, we present an embedding algorithm for graphs with maximum degree three that uses an n x n grid and a more complex algorithm for embedding a graph with maximum degree four. We also show a variation using orthogonal edges for maximum degree four graphs that also uses an n x n grid. The results have implications in graph theory, graph drawing, and VLSI design.