On Generalised Petersen Graphs of Girth 7 that have Cop Number 4
Harmony Morris, Joy Morris

TL;DR
This paper characterizes when generalized Petersen graphs with girth 7 have a cop number of 4, expanding understanding of their structural properties and confirming previous bounds for specific cases.
Contribution
It identifies exact conditions on n and k for girth 7 Petersen graphs to have cop number 4, explaining previously unclassified cases and situating them within infinite families.
Findings
Cop number is 4 when n=7k/i with i in {1,2,3}.
Provides a complete classification for girth 7 generalized Petersen graphs with cop number 4.
Extends previous results that only covered girth 8 cases.
Abstract
We show that if with then the cop number of the generalised Petersen graph is , with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most . The results in this paper explain all of the known generalised Petersen graphs that actually have cop number but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number and girth , while this paper explains those that have girth .)
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