# A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity

• Thomas Erlebach
• Jakob T. Spooner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)

## Abstract

This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph G. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then G is called cop-win, and robber-win otherwise. In contrast to previous work, we study this classical combinatorial game on edge-periodic graphs. These are graphs with an infinite lifetime comprised of discrete time steps such that each edge e is assigned a bit pattern of length $$l_e$$, with a 1 in the i-th position of the pattern indicating the presence of edge e in the i-th step of each consecutive block of $$l_e$$ steps. Utilising the known framework of reachability games, we obtain an $$O(\textsf {LCM}(L)\cdot n^3)$$ time algorithm to decide if a given n-vertex edge-periodic graph $$G^\tau$$ is cop-win or robber-win as well as compute a strategy for the winning player (here, L is the set of all edge pattern lengths $$l_e$$, and $$\textsf {LCM}(L)$$ denotes the least common multiple of the set L). For the special case of edge-periodic cycles, we prove an upper bound of $$2\cdot l \cdot \textsf {LCM}(L)$$ on the minimum length required of any edge-periodic cycle to ensure that it is robber-win, where $$l = 1$$ if $$\textsf {LCM}(L) \ge 2\cdot \max L$$, and $$l=2$$ otherwise. Furthermore, we provide constructions of edge-periodic cycles that are cop-win and have length $$1.5 \cdot \textsf {LCM}(L)$$ in the $$l=1$$ case and length $$3\cdot \textsf {LCM}(L)$$ in the $$l=2$$ case.

## Notes

### Acknowledgements

The authors would like to thank Maciej Gazda for helpful discussions regarding reachability games, as well as an anonymous reviewer for a suggestion leading to the running-time for the variant with k cops mentioned at the end of Sect. 3.

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## Authors and Affiliations

1. 1.School of InformaticsUniversity of LeicesterLeicesterEngland