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Depth-First Search and Kuratowski Subgraphs S. G. W I L L I A M S O N Untverstty o/Californta, San Dtego, La Jolla, California Abstract. Lel G = (V, E) be a nonplanar graph. The method of using depth-first techniques to extract a Kuratowski .,;ubgraph in time O( I V[ ) is shown. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems--complextty of proof procedures; computations on discrete structures General Terms: Algorithms Additional Key Words and Phrases: Nonplanar graphs, Kuratowski subgraphs, linear time algorithms 1. Introduction A number of efficient graph algorithms use the general method of depth-first search [3]. One of the deepest and most interesting of the depth-first search algorithms is the linear time planarity test of Hoperofi and Tarjan [2]. A classical result of Kuratowski (Theorem 1 below) states that, if G = (V, E) is nonplanar, then G contains a subgraph that is essentially (i.e., homeomorphic to) 1£3,3 (the complete bipartite graph based on the partition 3, 3) or K5 (the complete graph on 5 vertices). The basic depth-first search planarity test is not based on an explicit attempt to construct these residual nonplanar structures. In this paper
Journal of the ACM (JACM) – Association for Computing Machinery
Published: Sep 20, 1984
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