Tight Bounds on the Size of 2-Monopolies
نویسندگان
چکیده
This paper deals with the question of the innuence of a monopoly of vertices, seeking to gain the majority in local neighborhoods in a graph. Say that a vertex v is r-controlled by a set of vertices M if the majority of its neighbors at distance r are from M. We ask how large must M be in order to r-monopolize the graph, namely, r-control every vertex. Tight upper and lower bounds are provided for this problem, establishing that in an n-vertex graph, an r-monopoly M (for any even r 2) must be of size (n 3=5), and that for any r 2 there exist n-vertex graphs with r-monopolies of size O(n 3=5). This settles a problem left open in LPRS93, BePe95].
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