On a Generalization of Bi-Complement Reducible Graphs
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: Star1,2,3, Sun4 and P7, where Star1,2,3 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 Star1,2,3 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.
KeywordsShort Path Bipartite Graph Steiner Tree Free Graph Vertex Label
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