On algorithms for (P5,gem)-free graphs

Authors:
Hans L. Bodlaender;Andreas Brandstädt;Dieter Kratsch;Michaël Rao;Jeremy Spinrad
Affiliations:
Institute for Information and Computing Sciences, Utrecht University, TB Utrecht, The Netherlands;Fachbereich Informatik, Universität Rostock, Rostock, Germany;Université de Metz, Laboratoire d'Informatique Théorique et Appliquée, Metz Cedex, France;Université de Metz, Laboratoire d'Informatique Théorique et Appliquée, Metz Cedex, France;Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN
Venue:
Theoretical Computer Science - Graph colorings
Year:
2005

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Abstract

A graph is (P5, gem)-free, when it does not contain P5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P4) as an induced subgraph.We present O(n2) time recognition algorithms for chordal gem-free graphs and for(P5, gem)-free graphs. Using a characterization of (P5, gem)-free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees [A. Brandstädt, D. Kratsch, On the structure of (P5, gem)-free graphs, Discrete Appl. Math. 145 (2005), 155-166], we give linear time algorithms for the following NP-complete problems on (P5, gem)-free graphs: Minimum Coloring; Maximum Weight Stable Set; Maximum Weight Clique; and Minimum Clique Cover.