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Results on Independent Sets in Categorical Products of Graphs, the Ultimate Categorical Independence Ratio and the Ultimate Categorical Independent Domination Ratio

  • Wing-Kai Hon
  • Ton Kloks
  • Ching-Hao Liu
  • Hsiang-Hsuan Liu
  • Sheung-Hung Poon
  • Yue-Li Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)

Abstract

We first present polynomial algorithms to compute maximum independent sets in the categorical products of two cographs or two splitgraphs, respectively. Then we prove that computing the independent set of the categorical product of a planar graph of maximal degree three and K 4 is NP-complete. The ultimate categorical independence ratio of a graph G is defined as lim k → ∞  α(G k )/n k . The ultimate categorical independence ratio can be computed in polynomial time for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. Also, we present an O  ∗ (3 n/3) exact, exponential algorithm for the ultimate categorical independence ratio of general graphs. We further present a PTAS for the ultimate categorical independence ratio of planar graphs. Lastly, we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).

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References

  1. 1.
    Albertson, M., Collins, K.: Homomorphisms of 3-chromatic graphs. Discrete Mathematics 54, 127–132 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Lubetzky, E.: Independent sets in tensor graph powers. Journal of Graph Theory 54, 73–87 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Aurenhammer, F., Hagauer, J., Imrich, W.: Cartesian graph factorization at logarithmic cost per edge. Computational Complexity 2, 331–349 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brown, J., Nowakowski, R., Rall, D.: The ultimate categorical independence ratio of a graph. SIAM Journal on Discrete Mathematics 9, 290–300 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chartrand, G., Kapoor, S.F., Lick, D.R., Schuster, S.: The partial complement of graphs. Periodica Mathematica Hungarica 16, 83–95 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Corneil, D., Perl, Y., Stuwart, L.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14, 926–934 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cunningham, W.: Computing the binding number of a graph. Discrete Applied Mathematics 27, 283–285 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Hahn, G., Hell, P., Poljak, S.: On the ultimate independence ratio of a graph. European Journal of Combinatorics 16, 253–261 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hahn, G., Tardif, C.: Graph homomorphisms: structure and symmetry. In: Graph Symmetry – Algebraic Methods and Applications. NATO ASI Series C: Mathematical and Physical Sciences, vol. 497, pp. 107–166. Kluwer (1997)Google Scholar
  10. 10.
    Hedetniemi, S.: Homomorphisms of graphs and automata. Technical report 03105-44-T, University of Michigan (1966)Google Scholar
  11. 11.
    Hell, P., Nešetřil, J.: Graphs and homomorphisms. Oxford Univ. Press (2004)Google Scholar
  12. 12.
    Hell, P., Yu, X., Zhou, H.: Independence ratios of graph powers. Discrete Mathematics 27, 213–220 (1994)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Jha, P., Klavžar, S.: Independence in direct-product graphs. Ars Combinatoria 50, 53–63 (1998)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Kloks, T., Lee, C., Liu, J.: Stickiness, edge-thickness, and clique-thickness in graphs. Journal of Information Science and Engineering 20, 207–217 (2004)MathSciNetGoogle Scholar
  15. 15.
    Kloks, T., Wang, Y.: Advances in graph algorithms (Manuscript 2013)Google Scholar
  16. 16.
    Lubetzky, E.: Graph powers and related extremal problems. PhD Thesis, Tel Aviv University, Israel (2007)Google Scholar
  17. 17.
    Moon, J., Moser, L.: On cliques in graphs. Israel Journal of Mathematics 3, 23–28 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ravindra, G., Parthasarathy, K.: Perfect product graphs. Discrete Mathematics 20, 177–186 (1977)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Tóth, Á.: Answer to a question of Alon and Lubetzky about the ultimate categorical independence ratio. Manuscript on arXiv:1112.6172v1 (2011)Google Scholar
  20. 20.
    Tóth, Á.: The ultimate categorical independence ratio of complet multipartite graphs. SIAM Journal on Discrete Mathematics 23, 1900–1904 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM Journal on Computing 6, 505–517 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Weichsel, P.: The Kronecker product of graphs. Proceedings of the American mathematical Society 13, 47–52 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Woodall, D.: The binding number of a graph and its Anderson number. Journal of Combinatorial Theory, Series B 15, 225–255 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Zhang, H.: Independent sets in direct products of vertex-transitive graphs. Journal of Combinatorial Theory, Series B 102, 832–838 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Zhu, X.: The fractional version of Hedetniemi’s conjecture is true. European Journal of Combinatorics 32, 1168–1175 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Wing-Kai Hon
    • 1
  • Ton Kloks
    • 1
  • Ching-Hao Liu
    • 1
  • Hsiang-Hsuan Liu
    • 1
  • Sheung-Hung Poon
    • 1
  • Yue-Li Wang
    • 2
  1. 1.Department of Computer ScienceNational Tsing Hua UniversityTaiwan
  2. 2.Department of Information ManagementNational Taiwan University of Science and TechnologyTaiwan

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