On Maximal Chain Subgraphs and Covers of Bipartite Graphs

  • Tiziana Calamoneri
  • Mattia Gastaldello
  • Arnaud Mary
  • Marie-France Sagot
  • Blerina Sinaimeri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem. The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time.

Keywords

Chain subgraph cover problem Enumeration algorithms Exact exponential algorithms 

References

  1. 1.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics. Springer-Verlag, Heidelberg (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Eschen, E.M., Sritharan, R.: The induced matching and chain subgraph cover problems for convex bipartite graphs. Theor. Comput. Sci. 381(1), 260–265 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chang-Wu, Y., Gen-Huey, C., Tze-Heng, M.: On the complexity of the k-chain subgraph cover problem. Theor. Comput. Sci. 205(1), 85–98 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dias, V.M.F., de Figueiredo, C.M.H., Szwarcfiter, J.L.: Generating bicliques of a graph in lexicographic order. Theor. Comput. Sci. 337(1–3), 240–248 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dias, V.M.H., de Figueiredo, C.M.H., Szwarcfiter, J.L.: On the generation of bicliques of a graph. Discrete Appl. Math. 155(14), 1826–1832 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24(6), 1278–1304 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fedor, V.: Fomin and Dieter Kratsch. Exact Exponential Algorithms. Springer-Verlag New York Inc, New York, NY, USA (2010)Google Scholar
  9. 9.
    Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21(3), 618–628 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. 3(1), 23–28 (1965)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Nor, I., Engelstädter, J., Duron, O., Reuter, M., Sagot, M.-F., Charlat, S.: On the genetic architecture of cytoplasmic incompatibility: inference from phenotypic data. Am. Nat. 182(1), E15–E24 (2013)CrossRefGoogle Scholar
  14. 14.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods 3(3), 351–358 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Tiziana Calamoneri
    • 1
  • Mattia Gastaldello
    • 1
    • 2
  • Arnaud Mary
    • 2
  • Marie-France Sagot
    • 2
  • Blerina Sinaimeri
    • 2
  1. 1.Sapienza University of RomeRomaItaly
  2. 2.INRIA and Université de Lyon, Université Lyon 1, LBBE, CNRS UMR558VilleurbanneFrance

Personalised recommendations