Self-clique graphs and matrix permutations

  • Authors:

  • Adrian Bondy;Guillermo Durán;Min Chih Lin;Jayme L. Szwarcfiter

  • Affiliations:

  • UFR de Mathématiques, Université Claude-Bernard Lyon 1, France and Equipe Combinatoire CNRS, Université Paris 6, France;Depo. de Ingeniería Industrial, Fac. de Ciencias Físicas y Matemáticas, Univ. de Chile, Santiago, Chile and Depo. de Computación, Fac. de Ciencias Exactas y Naturales, Univ. de ...;Departemento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina;Instituto de Matemática, NCE and COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brasil

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2003

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Abstract

The clique graph of a graph is the intersection graph of its (maximal) cliques. A graph is self-clique when it is isomorphic with its clique graph, and is clique-Helly when its cliques satisfy the Helly property. We prove that a graph is clique-Helly and self-clique if and only if it admits a quasi-symmetric clique matrix, that is, a clique matrix whose families of row and column vectors are identical. We also give a characterization of such graphs in terms of vertex-clique duality. We describe new classes of self-clique and 2-self-clique graphs. Further, we consider some problems on permuted matrices (matrices obtained by permuting the rows and-or columns of a given matrix). We prove that deciding whether a (0,1)-matrix admits a symmetric (quasi-symmetric) permuted matrix is graph (hypergraph) isomorphism complete. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 178–192, 2003