# Small Alliances in Graphs

• Rodolfo Carvajal
• Martín Matamala
• Ivan Rapaport
• Nicolas Schabanel
Conference paper

DOI: 10.1007/978-3-540-74456-6_21

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)
Cite this paper as:
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

## 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 → ∞.

## 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