Efficient identification of Web communities
Proceedings of the sixth ACM SIGKDD international conference on Knowledge discovery and data mining
Introduction to the Theory of Computation
Introduction to the Theory of Computation
Local majorities, coalitions and monopolies in graphs: a review
Theoretical Computer Science
Discrete Applied Mathematics - Special issue on international workshop on algorithms, combinatorics, and optimization in interconnection networks (IWACOIN '99)
Approximation Algorithms
On the complement graph and defensive k-alliances
Discrete Applied Mathematics
On dissemination thresholds in regular and irregular graph classes
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
The complexity of the bootstraping percolation and other problems
Theoretical Computer Science
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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(3n2 - 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 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→8.