Abstract
A kernel in a directed graph D(V,E) is a set S of vertices of D such that no two vertices in S are adjacent and for every vertex u in \(V\smallsetminus S\) there is a vertex v in S , such that (u,v) is an arc of D. The problem of existence of a kernel is NP-complete for a general digraph. In this paper we introduce the strong kernel problem of an undirected graph G and solve it in polynomial time for circulant graphs.
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Alzoubi, K.M., Wan, P.J., Frieder, O.: New Distributed Algorithm for Connected Dominating Set in Wireless Ad Hoc Networks. In: Proc. 35th Hawaii Int. Conf. System Sciences, pp. 1–7 (2002)
Armugam, S., Velammal, S.: Maximum size of a connected graph with given domination parameters. Ars Combin. 52, 221–227 (1992)
Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorthims and Applications. Springer, London (2000)
Berge, C., Duchet, P.: Recent problems and results about kernels in directed graphs. Discrete Mathematics 86, 27–31 (1990)
Chvàtal, V.: On the computational complexity of finding a kernel. Report No. CRM-300, Centre de Recherches Mathematiques, Universite de Montreal (1973)
Berge, C.: Graphs, vol. 6. North Holland Publishing Co., Amsterdam (1985)
Fraenkel, A.S.: Planar kernel and Grundy with d ≤ 3,d + ≤ 2,d − ≤ 2 are NP -complete. Discrete Applied Mathematics 3, 257–262 (1981)
Bars, J.-M.L.: Counter example of the 0 − 1 Law for fragments of existential second-order logic: an overview. The Bulletin of Symbolic Logic 6(1), 67–82 (2000)
Wu, J.: Extended Dominating-Set-Based Routing in Ad Hoc Wireless Networks with Unidirectional Links. IEEE Transactions on Parallel and Distributed Systems 13(9), 866–881 (2002)
Kswásnik, M.: The generalisation of Richardson theorem. Discussiones Math. IV, 11–14 (1981)
Richardson, M.: Solutions of irreflexive relations. Ann. of Math. 58, 573–590 (1953)
Szwarcfiter, J.L., Chaty, G.: Enumerating the kernels of a directed graph with no odd circuits. Inform. Process. Lett. 51, 149–153 (1994)
Van Leeuwen, J.: Having a Grundy-numbering is NP-complete. Report No. 207, Computer Science Dept., Pennsylvania State University, University Park, PA (1976)
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton University Press, Princeton (1944)
Xu, J.: Topological Structure and Analysis of Interconnection Networks. Kluwer Academic Publishers, Dordrecht (2001)
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Manuel, P., Rajasingh, I., Rajan, B., Punitha, J. (2009). Kernel in Oriented Circulant Graphs. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_39
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DOI: https://doi.org/10.1007/978-3-642-10217-2_39
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