Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T20:05:25.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Algorithms for recognizing bipartite-Helly and bipartite-conformal hypergraphs*, **

Published online by Cambridge University Press:  24 October 2011

Marina Groshaus  and
Jayme Luis Szwarcfiter
Marina Groshaus
Affiliation:
Universidad de Buenos Aires, Departamento de Computación, CONICET Buenos Aires, Argentina. groshaus@dc.uba.ar
Jayme Luis Szwarcfiter
Affiliation:
Universidade Federal do Rio de Janeiro, IM, COPPE, and NCE, Rio de Janeiro, Brazil; jayme@nce.ufrj.br
Get access

Abstract

A hypergraph is Helly if every family of hyperedges of it, formed by pairwise intersecting hyperedges, has a common vertex. We consider the concepts of bipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way as conformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal and bipartite-Helly hypergraphs are also dual. They are useful for characterizing biclique matrices and biclique graphs, that is, the incident biclique-vertex incidence matrix and the intersection graphs of the maximal bicliques of a graph, respectively. These concepts play a similar role for the bicliques of a graph, as do clique matrices and clique graphs, for the cliques of the graph. We describe polynomial time algorithms for recognizing bipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.

Keywords

Algorithmsbipartite graphsbiclique-Hellybiclique matricesclique matricesHelly propertyhypergraphs
Type
Research Article
Information
RAIRO - Operations Research , Volume 45 , Issue 3 , July 2011 , pp. 209 - 222
Copyright
© EDP Sciences, ROADEF, SMAI 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amilhastre, J., Vilarem, M.C. and Janssen, P., Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs. Discrete Appl. Math. 86 (1998) 125–144. Google Scholar
C. Berge, Hypergraphs. Elsevier Science Publishers (1989).
Dias, V.M.F., Figueiredo, C.M.H. and Szwarcfiter, J.L., Generating bicliques in lexicographic order. Theoret. Comput. Sci. 337 (2005) 240–248. Google Scholar
Gavril, F., Algorithms on circular-arc graphs. Networks 4 (1974) 357–369. Google Scholar
Gilmore, P.C. and Hoffman, A.J., A characterization of comparability graphs and of interval graphs. Canadian J. Math. 16 (1964) 539–548. Google Scholar
Groshaus, M. and Szwarcfiter, J.L., Biclique-Helly graphs. Graphs Combin. 23 (2007) 633–645. Google Scholar
Groshaus, M. and Szwarcfiter, J.L., On hereditary Helly classes of graphs. Discrete Math. Theor. Comput. Sci. 10 (2008) 71–78. Google Scholar
Groshaus, M. and Szwarcfiter, J.L., Biclique graphs and biclique matrices. J. Graph Theory 63 (2010) 1–16. Google Scholar
Larrión, F., Neumann-Lara, V., Pizaña, M.A. and Porter, T.D., A hierarchy of self-clique graphs. Discrete Mathematics 282 (2004) 193–208. Google Scholar
Tuza, Zs., Covering of graphs by complete bipartite subgraphs: complexity of {0,1} matrices. Combinatorica 4 (1984) 111–116.Google Scholar