Small Alliances in Graphs

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 1 4 (3n −2n+3) and it can be computed in time O(n). 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 rp(G) = IP{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| > n 2 ), then also rp(G) → 1 as n → ∞.