Cops and attacking robbers with cycle constraints

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph $G$, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by $cc\left(G\right)$. We give a sufficient condition for a triangle-free graph to have attacking cop number at most 2 and we characterise when outerplanar graphs have attacking cop number 2. We also prove that all bipartite planar graphs $G$ have $cc\left(G\right)\le 4$ and show this is tight by constructing a bipartite planar graph $G$ with $cc\left(G\right)=4$. Finally we construct 17 non-isomorphic graphs $H$ of order 58 with $cc\left(H\right)=6$ and $c\left(H\right)=3$. This provides the first example of a graph $H$ with $cc\left(H\right)-c\left(H\right)\ge 3$, extending work by Bonato et al. (2014). We conclude with a list of conjectures and open problems.