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arXiv · 1712.07728

Throttling for the game of Cops and Robbers on graphs

Jane Breen·Boris Brimkov·Joshua Carlson·Leslie Hogben·K. E. Perry·Carolyn Reinhart
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Abstract

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are $ O(\sqrt n)$.

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Jane Breen, Boris Brimkov, Joshua Carlson, Leslie Hogben, K. E. Perry, Carolyn Reinhart. 2017-12-20. Throttling for the game of Cops and Robbers on graphs. https://arxiv.org/abs/1712.07728

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