# Small Alliances in Graphs

• Rodolfo Carvajal
• Martín Matamala
• Ivan Rapaport
• Nicolas Schabanel
Conference paper
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 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 → ∞.

## Keywords

Minimum Degree Powerful Alliance Facial Cycle High Order Cluster Bridgeless Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## 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