Variations of cops and robber on the hypercube

We determine the cop number of the hypercube for different versions of the game Cops and Robber. Cops and Robber is a two player game played on an undirected graph. One player controls some number of cops; the other player controls a single robber. In the standard game, the cops first choose some vertices to occupy, then the robber chooses a vertex; the players then alternate turns. On a turn, the robber may stay still or move to an adjacent vertex, likewise for the cops. The cop number of a graph is the least number of cops needed to guarantee the robber will be caught. The n-dimensional hypercube (or n-cube) is the graph whose vertices are the length n binary vectors with an edge between vectors that differ at exactly one coordinate. Various authors have investigated the cop number of the n-cube for a few versions of the game. We extend the game rules by considering cops of varying capacities. We refer to a cop who must move as an active cop, and cops who may move or stay still as flexible cops. Three versions of the game have been considered: 1) All cops are flexible, 2) All cops are active, and 3) At least one cop is active. In addition to the active and flexible cops, we introduce a third kind of cop, a passive cop, which must remain still on a turn. By varying the number of flexible, active, and passive cops we consider a whole spectrum of games. We fully classify the tradeoff between active and flexible cops. Introducing passive cops dramatically increases the difficulty and interest of the problem. The most involved proof of the paper achieves only a partial result for the game involving passive cops, and suggests a number of open questions. Finally, we connect Cops and Robber to another vertex pursuit game, Graph Searching, and use this relationship to provide a new lower bound for the cop number of the Graph Searching game.