Offensive r-alliances in graphs

  • Authors:

  • Henning Fernau;Juan A. Rodríguez;José M. Sigarreta

  • Affiliations:

  • FB 4-Abteilung Informatik, Universität Trier, 54286 Trier, Germany;Departament d'Enginyeria Informítica i Matemítiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain;Departamento de Matemáticas, Universidad Autónoma Popular del Estado de Puebla, 21 Sur 1103, Col. Santiago, Puebla, Mexico

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

Let G=(V,E) be a simple graph. For a nonempty set X@?V, and a vertex v@?V,@d"X(v) denotes the number of neighbors v has in X. A nonempty set S@?V is an offensiver-alliance in G if @d"S(v)=@d"S"@?(v)+r,@?v@?@?(S), where @?(S) denotes the boundary of S. An offensive r-alliance S is called global if it forms a dominating set. The global offensiver-alliance number of G, denoted by @c"r^o(G), is the minimum cardinality of a global offensive r-alliance in G. We show that the problem of finding optimal (global) offensive r-alliances is NP-complete and we obtain several tight bounds on @c"r^o(G).