Small Alliances in Graphs

  • Rodolfo Carvajal
  • Martín Matamala
  • Ivan Rapaport
  • Nicolas Schabanel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

Let G = (V,E) be a graph. A nonempty subset S ⊆ V is a (strong defensive) alliance of G if every node in S has at least as many neighbors in S than in V ∖ S. This work is motivated by the following observation: when G is a locally structured graph its nodes typically belong to small alliances. Despite the fact that finding the smallest alliance in a graph is NP-hard, we can at least compute in polynomial time depthG(v), the minimum distance one has to move away from an arbitrary node v in order to find an alliance containing v.

We define depth(G) as the sum of depthG(v) taken over v ∈ V. We prove that depth(G) can be at most \(\frac{1}{4}(3n^2-2n+3)\) and it can be computed in time O(n3). Intuitively, the value depth(G) should be small for clustered graphs. This is the case for the plane grid, which has a depth of 2n. We generalize the previous for bridgeless planar regular graphs of degree 3 and 4.

The idea that clustered graphs are those having a lot of small alliances leads us to analyze the value of Open image in new window {S contains an alliance}, with S ⊆ V randomly chosen. This probability goes to 1 for planar regular graphs of degree 3 and 4. Finally, we generalize an already known result by proving that if the minimum degree of the graph is logarithmically lower bounded and if S is a large random set (roughly \(|S| > \frac{n}{2})\), then also rp(G) →1 as n → ∞.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Rodolfo Carvajal
    • 1
  • Martín Matamala
    • 1
    • 2
  • Ivan Rapaport
    • 1
    • 2
  • Nicolas Schabanel
    • 2
    • 3
  1. 1.Departamento de Ingeniería Matemática, Universidad deChile
  2. 2.Centro de Modelamiento Matemático, Universidad deChile
  3. 3.LIP, École Normale Supérieure de LyonFrance

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