Immunity against Local Influence

This paper considers the question of the influence of a coalition of vertices, seeking to gain control (or majority) in local neighborhoods in a graph. A vertex v is said to be controlled by the coalition M if the majority of its neighbors are from M. Let Ruled(G, M) denote the set of vertices controlled by M in G. Previous studies focused on constructions allowing small coalitions to control many vertices, and provided tight bounds for the maximum possible size of Ruled(G, M) (as a function of vertical bar M vertical bar). This paper introduces the dual problem, concerning the existence and construction of graphs immune to the influence of small coalitions, i.e., graphs G for which Ruled(G, M) is small (relative to vertical bar M vertical bar again) for every coalition M. Upper and lower bounds are derived on the extent to which such immunity can be achieved.