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 ( G ) . 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 ( G ) ≤ 4 and show this is tight by constructing a bipartite planar graph G with cc ( G ) = 4 . Finally we construct 17 non-isomorphic graphs H of order 58 with cc ( H ) = 6 and c ( H ) = 3 . This provides the first example of a graph H with cc ( H ) − c ( H ) ≥ 3 , extending work by Bonato et al. (2014). We conclude with a list of conjectures and open problems.

Abstract We consider two variants of orthogonal colouring games on graphs. In these games, two players alternate colouring uncoloured vertices (from a choice of $$m\in {\mathbb {N}}$$ m ∈ N colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the partial colourings. In the normal play variant, the first player unable to move loses. In the scoring variant, each player aims to maximise their score, which is the number of coloured vertices in their copy of the graph. We prove that, given an instance with partial colourings, both the normal play and the scoring variant of the game are PSPACE-complete. An involution $$\sigma $$ σ of a graph G is strictly matched if its fixed point set induces a clique and $$v\sigma (v)\in E(G)$$ v σ ( v ) ∈ E ( G ) for any non-fixed point $$v\in V(G)$$ v ∈ V ( G ) . Andres et al. (Theor Comput Sci 795:312–325, 2019) gave a solution of the normal play variant played on graphs that admit a strictly matched involution. We prove that recognising graphs that admit a strictly matched involution is NP-complete.