Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

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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 C₄-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 C₄-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(n²+αm) time and O(m) space. (Here n, m, and α= O(m¹/²) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C₄-dominated graphs that contain no triangles in O(αm) time and O(m) space. Finally, we show how to enumerate all the maximal bicliques of a C₄-dominated graph with no triangles in O(n² + αm) time and O(αm) space, and we discuss how some biclique problems can be solved in O(αm) time and O(n+m) space.