Clique-critical graphs: maximum size and recognition

  • Authors:

  • Liliana Alcón

  • Affiliations:

  • Departamento de Matemática, Universidad Nacional de La Plata, La Plata, Argentina

  • Venue:
  • Discrete Applied Mathematics - Special issue: Traces of the Latin American conference on combinatorics, graphs and applications: a selection of papers from LACGA 2004, Santiago, Chile
  • Year:
  • 2006

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Abstract

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K-1 (G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.