Abstract
Let G be a graph. The independence-domination number γ i(G) is the maximum over all independent sets I in G of the minimal number of vertices needed to dominate I. In this paper we investigate the computational complexity of γ i(G) for graphs in several graph classes related to cographs. We present an exact exponential algorithm. We show that there is a polynomial-time algorithm to compute a maximum independent set in the Cartesian product of two cographs. We prove that independence domination is NP-hard for planar graphs and we present a PTAS.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aharoni, R., Berger, E., Ziv, R.: A tree version of Kőnig’s theorem. Combinatorica 22, 335–343 (2002)
Aharoni, R., Szabó, T.: Vizing’s conjecture for chordal graphs. Discrete Mathematics 309, 1766–1768 (2009)
Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41, 153–180 (1994)
Baker, K., Fishburn, P., Roberts, F.: Partial orders of dimension 2. Networks 2, 11–28 (1971)
Bertossi, A.: Dominating sets for split and bipartite graphs. Information Processing Letters 19, 37–40 (1984)
Bodlaender, H.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209, 1–45 (1998)
Booth, K., Johnson, J.: Domination in chordal graphs. SIAM Journal on Computing 11, 191–199 (1982)
Corneil, D., Lerchs, H., Stewart-Burlingham, L.: Complement reducible graphs. Discrete Applied Mathematics 3, 163–174 (1981)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
Cygan, M., Pilipczuk, M., Pilipczuk, M.: Known algorithms for edge clique cover are probably optimal. Manuscript on ArXiV: 1203.1754v1 (2012)
Damiand, G., Habib, M., Paul, C.: A simple paradigm for graph recognition: application to cographs and distance hereditary graphs. Theoretical Computer Science 263, 99–111 (2001)
Domke, G., Fisher, D., Ryan, J., Majumdar, A.: Fractional domination of strong direct products. Discrete Applied Mathematics 50, 89–91 (1994)
Farber, M.: Domination, independent domination, and duality in strongly chordal graphs. Discrete Applied Mathematics 7, 115–136 (1984)
Fisher, D.: Domination, fractional domination, 2-packings, and graph products. SIAM Journal on Discrete Mathematics 7, 493–498 (1984)
Fomin, F., Kratsch, D.: Exact exponential algorithms. EATCS series, Texts in Theoretical Computer Science. Springer (2010)
Golumbic, M.: Algorithmic graph theory and perfect graphs. Annals of Discrete Mathematics, vol. 57. Elsevier (2004)
Gregory, D., Pullman, N.: On a clique covering problem of Orlin. Discrete Mathematics 41, 97–99 (1982)
Grötschel, M., Lovász, L., Schrijver, A.: Relaxations of vertex packing. Journal of Combinatorial Theory, Series B 40, 330–343 (1986)
Halin, R.: S-functions for graphs. Journal of Geometry 8, 171–186 (1976)
Hammack, R., Imrich, W., Klavzar, S.: Handbook of Product Graphs. CRC Press (2011)
Howorka, E.: A characterization of distance-hereditary graphs. The Quarterly Journal of Mathematics 28, 417–420 (1977)
Imrich, W., Klavžar, S.: Product graphs: structure and recognition. John Wiley & Sons, New York (2000)
Kloks, T.: Treewidth – Computations and Approximations. LNCS, vol. 842. Springer (1994)
Hung, L.-J., Kloks, T.: On some simple widths. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 204–215. Springer, Heidelberg (2010)
Kloks, T., Liu, C., Poon, S.: On edge-independent sets (2013) (manuscript)
Kloks, T., Wang, Y.: Advances in graph algorithms (2013) (Manuscript)
Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11, 329–343 (1982)
Lovász, L.: On the Shannon capacity of a graph. IEEE Transactions on Information Theory IT-25, 1–7 (1979)
Milanič, M.: A note on domination and independence-domination numbers of graphs. Ars Mathematica Contemporanea 6, 89–97 (2013)
Moon, J., Moser, L.: On cliques in graphs. Israel Journal of Mathematics 3, 23–28 (1965)
Oum, S.: Graphs of bounded rank-width, PhD Thesis, Princeton University (2005)
Scheinerman, E., Ullman, D.: Fractional graph theory. Wiley–Interscience, New York (1997)
Suen, S., Tarr, J.: An improved inequality related to Vizing’s conjecture. The Electronic Journal of Combinatorics 19, 8 (2012)
Tedder, M., Corneil, D., Habib, M., Paul, C.: Simpler linear-time modular decomposition via recursive factorizing permutations. Manuscript on ArXiv: 0710.3901 (2008)
Telle, J.: Vertex partitioning problems: characterization, complexity and algorithms on partial k-trees, PhD Thesis, University of Oregon (1994)
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)
Vizing, V.: Cartesian product of graphs. Vychisl. Sistemy, 209–212 (1963) (Russian)
Vizing, V.: Some unsolved problems in graph theory. Uspehi Mat. Nauk 23, 117–134 (1968) (Russian)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hon, WK., Kloks, T., Liu, HH., Poon, SH., Wang, YL. (2013). On Independence Domination. In: Gąsieniec, L., Wolter, F. (eds) Fundamentals of Computation Theory. FCT 2013. Lecture Notes in Computer Science, vol 8070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40164-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-40164-0_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40163-3
Online ISBN: 978-3-642-40164-0
eBook Packages: Computer ScienceComputer Science (R0)