Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

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 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(n 2 +αm) time and O(n+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(n + m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n 2 + αm) time and O(αm) space.Fil: Eguía, Martiniano. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Soulignac, Francisco Juan. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

On edge-sets of bicliques in graphs

A biclique is a maximal induced complete bipartite subgraph of a graph. We investigate the intersection structure of edge-sets of bicliques in a graph. Specifically, we study the associated edge-biclique hypergraph whose hyperedges are precisely the edge-sets of all bicliques. We characterize graphs whose edge-biclique hypergraph is conformal (i.e., it is the clique hypergraph of its 2-section) by means of a single forbidden induced obstruction, the triangular prism. Using this result, we characterize graphs whose edge-biclique hypergraph is Helly and provide a polynomial time recognition algorithm. We further study a hereditary version of this property and show that it also admits polynomial time recognition, and, in fact, is characterized by a finite set of forbidden induced subgraphs. We conclude by describing some interesting properties of the 2-section graph of the edge-biclique hypergraph.Comment: This version corrects an error in Theorem 11 found after the paper went into prin

On hereditary Helly classes of graphs

Graphs and Algorithm

Absolute reflexive retracts and absolute bipartite retracts

AbstractIt is a well-known phenomenon in the study of graph retractions that most results about absolute retracts in the class of bipartite (irreflexive) graphs have analogues about absolute retracts in the class of reflexive graphs, and vice versa. In this paper we make some observations that make the connection explicit. We develop four natural transformations between reflexive graphs and bipartite graphs which preserve the property of being an absolute retract, and allow us to derive results about absolute reflexive retracts from similar results about absolute bipartite retracts and conversely. Then we introduce generic notions that specialize to the appropriate concepts in both cases. This paves the way to a unified view of both theories, leading to absolute retracts of general (i.e., partially reflexive) graphs