Small alliances in graphs

Let G = (V, E) be a graph. A nonempty subset S C 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 depth(G)(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 depth(G)(v) taken over v C V. We prove that depth(G) can be at most 1/4(3n(2) - 2n + 3) and it can be computed in time O(n(3)). 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 r(p)(G) = P{S contains an alliance}, with S subset of 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 vertical bar S vertical bar > 2), then also r(p)(G) -> 1 as n -> infinity.