Geometric graphs realization as coin graphs

Abstract

Koebe's Theorem [8] proves that any planar graph is the contact graph of a set of coins in the plane. But not any planar geometric graph can be realized as a coin graph (with coins centered at the vertices of the graph). This paper presents an algorithm to decide whether a planar connected geometric graph is a coin graph and to obtain, in the affirmative case, all the coin sets whose contact graphs are the given graph. This result is generalized to other metrics different from the Euclidean metric and is applied to a problem in mechanical gear systems. Two related optimization problems are also considered. They are motivated by graph drawing problems in Geographical Information Systems and Architectural Design Systems. © Springer-Verlag Berlin Heidelberg 2004.