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http://dx.doi.org/10.4134/JKMS.2008.45.3.871

CORE STABILITY OF DOMINATING SET GAMES  

Kong, Liang (Department of Mathematics Ocean University of China)
Fang, Qizhi (Department of Mathematics Ocean University of China)
Kim, Hye-Kyung (Department of Mathematics Catholic University of Daegu)
Publication Information
Journal of the Korean Mathematical Society / v.45, no.3, 2008 , pp. 871-881 More about this Journal
Abstract
In this paper, we study the core stability of the dominating set game which has arisen from the cost allocation problem related to domination problem on graphs. Let G be a graph whose neighborhood matrix is balanced. Applying duality theory of linear programming and graph theory, we prove that the dominating set game corresponding to G has the stable core if and only if every vertex belongs to a maximum 2-packing in G. We also show that for dominating set games corresponding to G, the core is stable if it is large, the game is extendable, or the game is exact. In fact, the core being large, the game being extendable and the game being exact are shown to be equivalent.
Keywords
dominating set game; balanced; stable core; largeness; exactness; extendability;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
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1 D. Fulkerson, A. Hoffman, and R. Oppenheim, On balanced matrices, Pivoting and extensions, Math. Programming Stud. No. 1 (1974), 120-132
2 T. Bietenhader and Y. Okamoto, Core Stability of Minimum Coloring Games, Graphtheoretic concepts in computer science, 389-401, Lecture Notes in Comput. Sci., 3353, Springer, Berlin, 2004
3 X. Deng, T. Ibaraki, and H. Nagamochi, Algorithmic aspects of the core of combinatorial optimization games, Math. Oper. Res. 24 (1999), no. 3, 751-766   DOI
4 X. Deng and C. H. Papadimitriou, On the complexity of cooperative solution concepts, Math. Oper. Res. 19 (1994), no. 2, 257-266   DOI   ScienceOn
5 M. Farber, Applications of LP Duality to Problems Involving Independence and Domination, Technical Report, Department of Computer Science, Simon Fraser University, Burnaby, Canada, 1981
6 K. Kikuta and L. S. Shapley, Core Stability in n-person Games, Manuscript, 1986
7 H. K. Kim and Q. Fang, Balancedness of integer domination games, J. Korean Math. Soc. 43 (2006), no. 2, 297-309   과학기술학회마을   DOI   ScienceOn
8 A. Lubiw, $\Gamma$-free matrices, Masters Thesis, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada, 1982
9 L. S. Shapley, Cores of convex games, Internat. J. Game Theory 1 (1971/72), 11-26   DOI
10 W. W. Sharkey, Cooperative games with large cores, Internat. J. Game Theory 11 (1982), no. 3-4, 175-182   DOI
11 T. Solymosi and T. E. S Raghavan, Assignment games with stable core, Internat. J. Game Theory 30 (2001), no. 2, 177-185   DOI
12 B. van Velzen, Dominating set games, Oper. Res. Lett. 32 (2004), no. 6, 565-573   DOI   ScienceOn
13 J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944
14 J. R. G. van Gellekom, J. A. M. Potters, and J. H. Reijnierse, Prosperity properties of TU-games, Internat. J. Game Theory 28 (1999), no. 2, 211-227   DOI