# 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}, Sun_{4} and P_{7}, 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 Sun_{4} 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 Sun_{4} 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.

## Keywords

Short Path Bipartite Graph Steiner Tree Free Graph Vertex Label## Preview

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