Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

• Autores: Eguía, Martiniano; Soulignac, Francisco Juan
• Año de publicación: 2013
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: HEREDITARY BICLIQUE-HELLY GRAPHS; MAXIMAL BICLIQUES; TRIANGLE-FREE GRAPHS; DOMINATION PROBLEMS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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; Argentina
Fil: Soulignac, Francisco Juan. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina