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Small alliances in a weighted graph
Authors:Kenji Kimura
Affiliation:
  • a Department of Management Science, Tokyo University of Science, Japan
  • b Department of Computer Science, Nihon University, Sakurajosui 3-25-40, Setagaya-Ku, Tokyo 156-8550, Japan
  • Abstract:We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑xNG(v)∩Sw(x)+w(v)≥∑xNG(v)−Sw(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most View the MathML source. In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most View the MathML source. We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3].
    Keywords:Defensive alliance   Alliance   Weighted graph
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