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Biclique-Helly Graphs

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Abstract

A graph is biclique-Helly when its family of (maximal) bicliques is a Helly family. We describe characterizations for biclique-Helly graphs, leading to polynomial time recognition algorithms. In addition, we relate biclique-Helly graphs to the classes of clique-Helly, disk-Helly and neighborhood-Helly graphs.

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References

  • Bandelt, H.-J., Farber, M., Hell, P.: Absolute reflexive retracts and absolute bipartite graphs. Discrete Applied Mathematics 44, 9–20 (1993)

    Google Scholar 

  • Bandelt, H.-J., Pesch, E.: Dismantling absolute retracts of reflexive graphs. European J. Combinatorics 10, 211–220 (1989)

    Google Scholar 

  • Bandelt, H.-J., Prisner, E.: Clique graphs and Helly graphs. J. Combin Theory B 51, 34–45 (1991)

    Google Scholar 

  • Berge, C.: Hypergraphs. North Holland Mathematical Library, vol.45, Elsevier Science Publishers B.V., Amsterdam, 1989

  • Berge, C., Duchet, P.: A generalization of Gilmore’s theorem. In: Fiedler, M. (ed.) Recent Advances in Graph Theory. Acad. Praha, Prague pp 49–55 (1975)

  • Bondy, A., Durán, G., Lin, M., Szwarcfiter, J.: Self-clique graphs and matrix permutations. Journal of Graph Theory 44, 178–192 (2003)

    Google Scholar 

  • Burzyn, P., Bonomo, F., Durán, G.: NP-completeness results for edge modification problems. Discrete Applied Mathematics 154(13), 1824–1844 (2006)

    Google Scholar 

  • Dragan, F.F.: Centers of Graphs and the Helly property. PhD thesis, Moldava State University, Chisinau, Moldava (1989) (In russian)

  • Escalante, F.: Über iterierte Clique-Graphen. Abhandlungender Mathematischen Seminar der Universität Hamburg 39, 59–68 (1973)

  • Hamelink, B.C.: A partial characterization of clique graphs. J. Combin Theory 5, 192–197 (1968)

    Google Scholar 

  • Pavol Hell, Personal Communication (2003)

  • Larrión, F., Neumann-Lara, V., Pizaña, M.A., Porter, T.D.: A hierarchy of self-clique graphs. Discrete Mathematics 282, 193–208 (2004)

    Google Scholar 

  • Müller, H.: On edge perfectness and classes of bipartite graphs. Discrete Math. 149, 159–187 (1996)

    Google Scholar 

  • Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Appl. Math. 131, 651–654 (2003)

    Google Scholar 

  • Prisner, E.: Bicliques in graphs I. Bounds on their number. Combinatorica 20, 109–117 (2000)

    Google Scholar 

  • Prisner, E.: Bicliques in graphs II. Recognizing k-path graphs and underlying graphs of line digraphs. Graph Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 1335, 253–287 (1997)

  • Szwarcfiter, J.L.: Recognizing clique-Helly graphs. Ars Combinatoria 45, 29–32 (1997)

    Google Scholar 

  • Tuza, Z.: Covering of graphs by complete bipartite subgraphs: complexity of 0-1 matrices. Combinatorica 4, 111–116 (1984)

    Google Scholar 

  • Robert, F.S., Spencer, J.H.: A characterization of clique graphs. J. Combin. Theory B 10, 102–108 (1971)

    Google Scholar 

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Authors and Affiliations

  1. Facultad de Ciencias Exáctas y Naturales, Departamento de Computación, Universidad de Buenos Aires, Argentina

    Marina Groshaus

  2. Instituto de Matemática, NCE and COPPE, Universidade Federal do Rio de Janeiro, Brasil

    Jayme L. Szwarcfiter

Authors
  1. Marina Groshaus
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  2. Jayme L. Szwarcfiter
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Corresponding author

Correspondence to Jayme L. Szwarcfiter.

Additional information

Partially supported by UBACyT Grants X184, X127, X212, PICT ANPCyT Grant 11-09112 and PID Conicet Grant, Argentina.

Partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, Brasil.

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Groshaus, M., Szwarcfiter, J.L. Biclique-Helly Graphs. Graphs and Combinatorics 23, 633–645 (2007). https://doi.org/10.1007/s00373-007-0756-6

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  • DOI: https://doi.org/10.1007/s00373-007-0756-6

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