On Edge-Independent Sets

  • Ton Kloks
  • Ching-Hao Liu
  • Sheung-Hung Poon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7924)


A set of edges in a graph G is independent if no two elements are contained in a clique of G. The edge-independent set problem asks for the maximal cardinality of independent sets of edges. We show that the edge-clique graphs of cocktail parties have unbounded rankwidth. There is an elegant formula that solves the edge-independent set problem for graphs of rankwidth one, which are exactly distance-hereditary graphs, and related classes of graphs. We present a PTAS for the edge-independent set problem on planar graphs and show that the problem is polynomial for planar graphs without triangle separators. The set of edges of a bipartite graph is edge-independent. We show that the edge-independent set problem remains NP-complete for graphs in which every neighborhood is bipartite, i.e., the graphs without odd wheels.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albertson, M., Collins, K.: Duality and perfection for edges in cliques. Journal of Combinatorial Theory, Series B 36, 298–309 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alcón, L., Faria, L., de Figueiredo, C., Gutierrez, M.: The complexity of clique graph recognition. Theoretical Computer Science 410, 2072–2083 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Anand, P., Escuadro, H., Gera, R., Hartke, S., Stolee, D.: On the hardness of recognizing triangular line graphs. Discrete Mathematics 312, 2627–2638 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41, 153–180 (1994)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bandelt, H., Mulder, H.: Distance-hereditary graphs. Journal of Combinatorial Theory, Series B 41, 182–208 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brouwer, A., Rees, G.: More mutually orthogonal Latin squares. Discrete Mathematics 39, 181–263 (1982)CrossRefGoogle Scholar
  7. 7.
    Cerioli, M.: Clique graphs and edge-clique graphs. Electronic Notes in Discrete Mathematics 13, 34–37 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cerioli, M., Szwarcfiter, J.: A characterization of edge clique graphs. Ars Combinatorica 60, 287–292 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cerioli, M., Szwarcfiter, J.: Edge clique graphs and some classes of chordal graphs. Discrete Mathematics 242, 31–39 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chowla, S., Erdös, P., Straus, E.: On the maximal number of pairwise orthogonal Latin squares of a given order. Canadian Journal of Mathematics 12, 204–208 (1960)zbMATHCrossRefGoogle Scholar
  11. 11.
    Corneil, D., Perl, Y., Stewart, L.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14, 926–934 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cygan, M., Pilipczuk, M., Pilipczuk, M.: Known algorithms for edge clique cover are probably optimal. In: Proceedings SODA 2013, ACM-SIAM, pp. 1044–1053 (2013)Google Scholar
  13. 13.
    Dvořák, Z., Král, D.: Classes of graphs with small rank decompositions are χ-bounded. European Journal of Combinatorics 33, 679–683 (2012)zbMATHCrossRefGoogle Scholar
  14. 14.
    Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Ganian, R., Hliněný, P.: Better polynomial algorithms on graphs of bounded rank-width. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 266–277. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Gao, J., Kloks, T., Poon, S.-H.: Triangle-partitioning edges of planar graphs, toroidal graphs and k-planar graphs. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 194–205. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    Gargano, L., Körner, J., Vaccaro, U.: Sperner capacities. Graphs and Combinatorics 9, 31–46 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gregory, D.A., Pullman, N.J.: On a clique covering problem of Orlin. Discrete Mathematics 41, 97–99 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gyárfás, A.: A simple lower bound on edge covering by cliques. Discrete Mathematics 85, 103–104 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Haxell, P., Kostoschka, A., Thomassé, S.: A stability theorem on fractional covering of triangles by edges. European Journal of Combinatorics 33, 799–806 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Howork, E.: A characterization of distance-hereditary graphs. The Quarterly Journal of Mathematics 28, 417–420 (1977)CrossRefGoogle Scholar
  22. 22.
    Jamison, B., Olariu, S.: A unique tree representation for P 4-sparse graphs. Discrete Applied Mathematics 35, 115–129 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Kong, J., Wu, Y.: On economical set representations of graphs. Discrete Mathematics and Theoretical Computer Science 11, 71–96 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Körner, J.: Intersection number and capacities of graphs. Discrete Mathematics 142, 169–184 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ma, F., Zhang, J.: Finding orthogonal Latin squares using finite model searching tools. Science China Information Sciences 56, 1–9 (2013)Google Scholar
  26. 26.
    Lakshmanan, S., Bujtás, C., Tuza, Z.: Small edge sets meeting all triangles of a graph. Graphs and Combinatorics 28, 381–392 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lakshmanan, S., Vijayakumar, A.: Clique irreducibility of some iterative classes of graphs. Discussiones Mathematicae Graph Theory 28, 307–321 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Le, V.B.: Gallai graphs and anti-Gallai graphs. Discrete Mathematics 159, 179–189 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Mujuni, E., Rosamond, F.: Parameterized complexity of the clique partition problem. In: Harland, J., Manyem, P. (eds.) Proceedings CATS 2008. ACS, CRPIT series, vol. 77, pp. 75–78 (2008)Google Scholar
  30. 30.
    Oum, S.: Graphs of bounded rankwidth. PhD thesis, Princeton University (2005)Google Scholar
  31. 31.
    Park, B., Kim, S., Sano, Y.: The competition numbers of complete multipartite graphs and mutually orthogonal Latin squares. Discrete Mathematics 309, 6464–6469 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Prisner, E.: Graph dynamics. Pitman Research Notes in Mathematics Series. Longman, Essex (1995)zbMATHGoogle Scholar
  33. 33.
    Raychaudhuri, A.: Intersection number and edge clique graphs of chordal and strongly chordal graphs. Congressus Numerantium 67, 197–204 (1988)MathSciNetGoogle Scholar
  34. 34.
    Raychaudhuri, A.: Edge clique graphs of some important classes of graphs. Ars Combinatoria 32, 269–278 (1991)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Wilson, R.: Concerning the number of mutually orthogonal Latin squares. Discrete Mathematics 9, 181–198 (1974)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ton Kloks
    • 1
  • Ching-Hao Liu
    • 1
  • Sheung-Hung Poon
    • 1
  1. 1.Department of Computer ScienceNational Tsing Hua UniversityTaiwan

Personalised recommendations