The power of small coalitions in graphs

This paper considers the question of the influence of a coalition of vertices, seeking to gain control (or majority) in local neighborhoods in a general graph. Say that a vertex nu is controlled by the coalition M if the majority of its neighbors are from M. We ask how many vertices (as a function of \M\) 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 fixed radius r greater than or equal to 1, or all neighborhoods of radii up to r). In particular, we look also at the case where the coalition must control all vertices (including or excluding its own), and derive bounds for its size. (C) 2002 Elsevier Science B.V. All rights reserved.