Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

Abstract

A biclique is a set of vertices that induce a bipartite complete graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C_4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C_4-dominated graphs that contain no triangles and no induced cycles of length either 5, or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n^2+\alpha m) time and O(m) space. (Here n, m, and \alpha = O(m^{1/2}) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C_4-dominated graphs that contain no triangles in O(\alpha m) time and O(m) space. Finally, we show how to enumerate all the maximal bicliques of a C_4-dominated graph with no triangles in O(n^2 + \alpha m) time and O(\alpha m) space, and we discuss how some biclique problems can be solved in O(\alpha m) time and O(n+m) space.