Computing global offensive alliances in Cartesian product graphs

  • Authors:

  • Ismael G. Yero;Juan A. RodríGuez-VeláZquez

  • Affiliations:

  • Departamento de Matemáticas, Escuela Politécnica Superior de Algeciras, Universidad de Cádiz, Av. Ramón Puyol s/n, 11202 Algeciras, Spain;Departament d'Enginyeria Informítica i Matemítiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

A global offensive alliance in a graph G is a set S of vertices with the property that every vertex not belonging to S has at least one more neighbor in S than it has outside of S. The global offensive alliance number of G, @c"o(G), is the minimum cardinality of a global offensive alliance in G. A set S of vertices of a graph G is a dominating set for G if every vertex not belonging to S has at least one neighbor in S. The domination number of G, @c(G), is the minimum cardinality of a dominating set of G. In this work we obtain closed formulas for the global offensive alliance number of several families of Cartesian product graphs, we also prove that @c"o(G@?H)=@c(G)@c"o(H)2 for any graphs G and H and we show that if G has an efficient dominating set, then @c"o(G@?H)=@c(G)@c"o(H). Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.