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Algorithms for perfectly contractile graphs

By Frédéric Maffray and Nicolas Trotignon

Abstract

International audienceWe consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class A0 of graphs that contain no odd hole, no antihole of length at least 5, and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are relevant to the study of perfect graphs. We give polynomial-time recognition algorithms for these two classes. We proved previously that every graph G in A is "perfectly contractile", as conjectured by Everett and Reed [see the chapter "Even pairs" in the book Perfect Graphs, J.L. Ramirez-Alfonsin and B.A. Reed, eds., Wiley Interscience, 2001]. The analogous conjecture concerning graphs in A0 is still open

Topics: perfect graph, perfectly contractile graph, recognition, coloring, AMS 05C17 05C85 68R10, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
Publisher: 'Society for Industrial & Applied Mathematics (SIAM)'
Year: 2005
DOI identifier: 10.1137/S0895480104442522
OAI identifier: oai:HAL:hal-00166911v1
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