### Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

*Martiniano Roberto Eguía, Francisco Juan Soulignac*

#### Abstract

A biclique is a set of vertices that induce a complete bipartite
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}+αm) time and O(n+m) space. (Here n, m, and α= 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(αm) time and O(n+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}+ αm) time and O(αm) space.Full Text: PDF PostScript