Bipartite dimensions and bipartite degrees of graphs Original Research Article

A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover Gʹs edges. Gʹsbipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d(G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d ⩽ 2, and investigate the forbidden graphs for d ⩽ n that have the fewest vertices. We note that for complete graphs, d(Kn) = [log2n], η(Kn) = d(Kn) for n ⩽ 16, and η(Kn) is unbounded. The list of minimal forbidden induced subgraphs for η ⩽ 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices.