Abstract
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 |M|). 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 |M| again) for every coalition M. Upper and lower bounds are derived on the extent to which such immunity can be achieved.
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Peleg, D. (2014). Immunity against Local Influence. In: Dershowitz, N., Nissan, E. (eds) Language, Culture, Computation. Computing - Theory and Technology. Lecture Notes in Computer Science, vol 8001. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45321-2_8
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