Abstract
Motivated by a biological scenario illustrated in the YouTube video \url{ this https URL} where a neutrophil chases a bacteria cell moving in random directions, we present a variant of the cop and robber game on graphs called the tipsy cop and drunken robber game. In this game, we place a tipsy cop and a drunken robber at different vertices of a finite connected graph $G$. The game consists of independent moves where the robber begins the game by moving to an adjacent vertex from where he began, this is then followed by the cop moving to an adjacent vertex from where she began. Since the robber is inebriated, he takes random walks on the graph, while the cop being tipsy means that her movements are sometimes random and sometimes intentional. Our main results give formulas for the probability that the robber is still free from capture after $m$ moves of this game on highly symmetric graphs, such as the complete graphs, complete bipartite graphs, and cycle graphs. We also give the expected encounter time between the cop and robber for these families of graphs. We end the manuscript by presenting a general method for computing such probabilities and also detail a variety of directions for future research.
Highlights
Motivated by a biological scenario where a neutrophil chases a bacteria cell moving in random directions, we present a variant of the cop and robber game on graphs called the tipsy cop and drunken robber game
We assume that the robber always moves first, and the robber’s movement takes place on odd numbered moves of the game and the cop on the even numbered moves
We tackle the following question in this paper: What is the probability that the robber is still free from capture after m moves of this game? We show that this probability depends only on the starting position of the cop and robber, and we give recursive formulas for these probabilities for complete graphs (Section 2) and complete bipartite graphs (Section 3)
Summary
Throughout every section that follows we let θ ∈ [0, 1] denote the proportion of the cop’s movements that are random. Since the probability that the robber is captured after m moves is Pm−1(G) − Pm(G), the expected encounter time is. Let X denote the random variable giving the capture time on the complete graph Kv. If θ denotes the proportion of the cop’s movement that is random, the expected encounter time between the robber and cop is given by. Let X denote the random variable giving the capture time in number of rounds (where a round consists of one robber move and one cop move) on the complete graph Kv. If θ denotes the proportion of the cop’s movement that is random, the expected encounter time between the robber and the cop is given by. It can be concluded that the expected time it would take the cop to catch the robber is given by the reciprocal of P
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