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 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 ∈ 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(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 
{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 r
p
(G) →1 as n → ∞.
Partially supported by Programs Conicyt “Anillo en Redes”, Fondap on Applied Mathematics and Ecos-Conicyt.
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Carvajal, R., Matamala, M., Rapaport, I., Schabanel, N. (2007). Small Alliances in Graphs. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_21
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