Recognizing Planar Perfect Graphs

Abstract

An O(n3) algorithm for recognizing planar graphs that do not contain induced odd cycles of length greater than 3 (odd holes) is presented. A planar graph with this property satisfies the requirement that its maximum clique size equal the minimum number of colors required for the graph (graphs all of whose induced subgraphs satisfy the latter property are perfect as defined by Berge). The algorithm presented is based on decomposing these graphs into essentially two special classes of inseparable component graphs that are easy to recognize. They are (i) planar comparability graphs and (ii) planar line graphs of those planar bipartite graphs whose maximum degrees are no greater than 3. Composition schemes for generating planar perfect graphs from those basic components are also provided. This decomposition algorithm can also be adapted to solve the corresponding maximum independent set and minimum coloring problems. Finally, the path-parity problem on planar perfect graphs is considered. © 1987, ACM. All rights reserved.