On biclique coverings

Abstract

It was proved by Fronček, Jerebic, Klavžar, and Kovář that if a complete bipartite graph K n,n with a perfect matching removed can be covered by k bicliques, then n k k 2 . We give a slightly simplified proof and we show that the result is tight. Moreover, we use the result to prove analogous bounds for coverings of some other classes of graphs by bicliques.

A set coverage problem

“…De Caen, Gregory and Pullman [2] proved that br(Ī n ) = s(n), whereĪ n is the complement of the identity matrix I n , and s(n) is the smallest integer k such that n k k 2 . The same result was recently rediscovered in [1,5] (see previous paragraph), with a new proof, for biclique covering of the bipartite graph K − n,n , where K − n,n is the complete bipartite graph K n,n with a perfect matching removed.…”

Some new bounds for cover-free families through biclique covers

“…So if we denote by s(t), the size of minimum weakly separating system, then we have s(t) = bc(K t ). Also, in [3], it was proved that…”

“…The number of bicliques in a minimum biclique cover of G is called the biclique covering number of G and denoted by bc(G). This measure of graphs is studied in the literature [1,3,15].…”

Computing Cross Associations for Attack Graphs and Other Applications

“…The biclique edge covering problem asks for the minimum number of bicliques to cover the edges of G. Results for a special case of the biclique edge covering problem are described by Bezrukov, et.al. [10]. Szeider [14] defines a total biclique cover as a collection of disjoint bicliques such that every vertex in the set A is in one of the bicliques.…”