Faster recognition of clique-Helly and hereditary clique-Helly graphs

Authors:
Min Chih Lin;Jayme L. Szwarcfiter
Affiliations:
Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Computación, Buenos Aires, Argentina;Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brazil
Venue:
Information Processing Letters
Year:
2007

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Abstract

A family of subsets of a set is Helly when every subfamily of it, which is formed by pairwise intersecting subsets contains a common element. A graph G is clique-Helly when the family of its (maximal) cliques is Helly, while G is hereditary clique-Helly when every induced subgraph of it is clique-Helly. The best algorithms currently known to recognize clique-Helly and hereditary clique-Helly graphs have complexities O(nm^2) and O(n^2m), respectively, for a graph with n vertices and m edges. In this Note, we describe algorithms which recognize both classes in O(m^2) time. These algorithms also reduce the complexity of recognizing some other classes, as disk-Helly graphs.