On a Generalization of Bi-Complement Reducible Graphs

Abstract

A graph is called complement reducible (a cograph for short) if every its induced subgraph with at least two vertices is either disconnected or the complement to a disconnected graph. The bipartite analog of cographs, bi-complement reducible graphs, has been characterized recently by three forbidden induced subgraphs: Star_1,2,3, Sun₄ and P₇, where Star_1,2,3 is the graph with vertices a,b,c,d,e, f,g and edges (a, b), (b,c), (c,d), (d,e), (e, f), (d,g), and Sun₄ is the graph with vertices a,b,c,d,e,f,g,h and edges (a,b), (b,c), (c,d), (d,a), (a, e), (b,f), (c, g), (d, h). In the present paper, we propose a structural characterization for the class of bipartite graphs containing no graphs Star_1,2,3 and Sun₄ as induced subgraphs. Based on the proposed characterization we prove that the clique-width of these graphs is at most five that leads to polynomial algorithms for a number of problems which are NP-complete in general bipartite graphs.

This research has been supported by the Russian Foundation for Basic Research (Grant 00-01-00601). Part of the study has been done while the author was visiting RUTCOR, Rutgers Center for Operations Research, Rutgers University. The support of the Office of Naval Research (Grant N00014-92-J-1375) and the National Science Foundation (Grant DMS-9806389) is gratefully acknowledged.