The complexity of clique graph recognition

Authors:
Liliana Alcón;Luerbio Faria;Celina M. H. de Figueiredo;Marisa Gutierrez
Affiliations:
Departamento de Matemática, UNLP, Argentina;Departamento de Matemática, FFP, UERJ, Brazil;Programa de Engenharia de Sistemas e Computação, COPPE, UFRJ, Brazil;CONICET, Departamento de Matemática, UNLP, Argentina
Venue:
Theoretical Computer Science
Year:
2009

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Abstract

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C(G) the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of C(G). Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. However, the time complexity of the problem of recognizing clique graphs is a long-standing open question. We prove that the clique graph recognition problem is NP-complete.