Abstract
An upper dominating set in a graph is a minimal dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in finitely defined classes of graphs and conjecture that the problem admits a complexity dichotomy in this family. A helpful tool to study the complexity of an algorithmic problem is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. We discover the first boundary class for this problem and prove the dichotomy for monogenic classes.
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AbouEisha, H., Hussain, S., Lozin, V., Monnot, J., Ries, B.: A dichotomy for upper domination in monogenic classes. Lecture Notes Comput. Sci. 8881, 258–267 (2014)
AbouEisha, H., Hussain, S., Lozin, V., Monnot, J., Ries, B., Zamaraev, V.: A boundary property for upper domination. Lecture Notes Comput. Sci. 9843, 229–240 (2016)
Alekseev, V.E.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discrete Appl. Math. 132, 17–26 (2003)
Alekseev, V.E., Korobitsyn, D.V., Lozin, V.V.: Boundary classes of graphs for the dominating set problem. Discrete Math. 285, 1–6 (2004)
Alekseev, V.E., Boliac, R., Korobitsyn, D.V., Lozin, V.V.: NP-hard graph problems and boundary classes of graphs. Theor. Comput. Sci. 389, 219–236 (2007)
Bazgan, C., Brankovic, L., Casel, K., Fernau, H., Jansen, K., Klein, K.-M., Lampis, M., Liedloff, M., Monnot, J., Paschos, VTh: Algorithmic aspects of upper domination: a parameterised perspective. Lecture Notes Comput. Sci. 9778, 113–124 (2016)
Bazgan, C., Brankovic, L., Casel, K., Fernau, H., Jansen, K., Klein, K.-M., Lampis, M., Liedloff, M., Monnot, J., Paschos, VTh: Upper domination: complexity and approximation. Lecture Notes Comput. Sci. 9843, 241–252 (2016)
Brandstädt, A., Engelfriet, J., Le, H.-O., Lozin, V.V.: Clique-width for 4-vertex forbidden subgraphs. Theory Comput. Syst. 39(4), 561–590 (2006)
Cheston, G.A., Fricke, G., Hedetniemi, S.T., Jacobs, D.P.: On the computational complexity of upper fractional domination. Discrete Appl. Math. 27(3), 195–207 (1990)
Cockayne, E.J., Favaron, O., Payan, C., Thomason, A.G.: Contributions to the theory of domination, independence and irredundance in graphs. Discrete Math. 33(3), 249–258 (1981)
Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)
Courcelle, B., Olariu, S.: Upper bounds to the clique-width of a graph. Discrete Appl. Math. 101, 77–114 (2000)
Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)
Hare, E.O., Hedetniemi, S.T., Laskar, R.C., Peters, K., Wimer, T.: Linear-time computability of combinatorial problems on generalized-series-parallel graphs. In: Johnson, D.S., et al. (eds.) Discrete Algorithms and Complexity, pp. 437–457. Academic Press, New York (1987)
Jacobson, M.S., Peters, K.: Chordal graphs and upper irredundance, upper domination and independence. Discrete Math. 86(1–3), 59–69 (1990)
Kamiński, M., Lozin, V., Milanič, M.: Recent developments on graphs of bounded clique-width. Discrete Appl. Math. 157, 2747–2761 (2009)
Korobitsyn, D.V.: On the complexity of determining the domination number in monogenic classes of graphs. Diskretnaya Matematika 2(3), 90–96 (1990) (in Russian, translation in Discrete Math. Appl. 2 (1992), no. 2, 191–199)
Korpelainen, N., Lozin, V.V., Malyshev, D.S., Tiskin, A.: Boundary properties of graphs for algorithmic graph problems. Theor. Comput. Sci. 412, 3545–3554 (2011)
Korpelainen, N., Lozin, V., Razgon, I.: Boundary properties of well-quasi-ordered sets of graphs. Order 30, 723–735 (2013)
Lozin, V.V.: Boundary classes of planar graphs. Comb. Probab. Comput. 17, 287–295 (2008)
Lozin, V., Milanič, M.: Critical properties of graphs of bounded clique-width. Discrete Math. 313, 1035–1044 (2013)
Lozin, V., Purcell, C.: Boundary properties of the satisfiability problems. Inf. Process. Lett. 113, 313–317 (2013)
Lozin, V., Rautenbach, D.: On the band-, tree- and clique-width of graphs with bounded vertex degree. SIAM J. Discrete Math. 18, 195–206 (2004)
Lozin, V., Zamaraev, V.: Boundary properties of factorial classes of graphs. J. Graph Theory 78, 207–218 (2015)
Lozin, V.V., Mosca, R.: Independent sets in extensions of \(2K_2\)-free graphs. Discrete Appl. Math. 146(1), 74–80 (2005)
Murphy, O.J.: Computing independent sets in graphs with large girth. Discrete Appl. Math. 35, 167–170 (1992)
Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Comb. Theory Ser. B. 41(1), 92–114 (1986)
Acknowledgements
Vadim Lozin and Viktor Zamaraev gratefully acknowledge support from EPSRC, Grant EP/L020408/1. Part of this research was carried out when Vadim Lozin was visiting the King Abdullah University of Science and Technology (KAUST). This author thanks the University for hospitality and stimulating research environment.
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AbouEisha, H., Hussain, S., Lozin, V. et al. Upper Domination: Towards a Dichotomy Through Boundary Properties. Algorithmica 80, 2799–2817 (2018). https://doi.org/10.1007/s00453-017-0346-9
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DOI: https://doi.org/10.1007/s00453-017-0346-9