Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs

Abstract

A biclique cover (resp. biclique decomposition) of a bipartite graph B is a family of complete bipartite subgraphs of B whose edges cover (resp. partition) the edges of B. The minimum cardinality of a biclique cover (resp. biclique decomposition) is denoted by s-dim( B) (resp. s-part( B)). The decision problems associated with the computation of s-dim and s-part are NP-complete for general bipartite graphs; the decision problem associated to s-dim is NP-complete for bipartite chordal graphs, and polynomial for bipartite distance-hereditary graphs, for bipartite convex graphs and for bipartite C4-free graphs. We show here that for bipartite domino-free graphs (a strict generalization of bipartite distance-hereditary graphs and bipartite C4-free graphs), s-dim and s-part are equal and can be computed in O(n × m) time. Moreover, we propose a O(n × m) time algorithm to check the domino-free property and to build the Galois lattice of such graphs.