The Power of Small Coalitions in Graphs

Description

This paper considers the question of the in uence of a coalition of vertices, seeking to gain control (or majority) in local neighborhoods in a general graph. Say that a vertex v is controlled by the coalition M if the majority of its neighbors are from M. We ask how many vertices (as a function of jMj) can M control in this fashion. Upper and lower bounds are provided for this problem, as well as for cases where the majority is computed over larger neighborhoods (either neighborhoods of some xed radius r 1, or all neighborhoods of radii up to r). In particular, we look also at the case where the coalition must control all vertices outside itself, and derive bounds for its size.

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