Cops that surround a robber
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We introduce the game of Surrounding Cops and Robber on a graph, as a variant of the traditional game of Cops and Robber. In contrast to the traditional game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber's neighbouring vertices. We denote by σ(G) the surrounding copnumber of G, namely the least number of cops required to surround a robber in the graph G. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest include product graphs, graphs arising from combinatorial designs, and generalised Petersen graphs.
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