12 problems

A game space is a compact geodesic metric space equipped with the Cops and Robber game. Its doubling constant is the least $k\in\mathbb{N}\cup\{\infty\}$ such that every ball…

Minimal-order conjecture for the attacking-cop-number gap. If

Strong attacking-cop-number conjecture. For all integers $k \geq 4$ there exists a graph $H$ such that $\operatorname{c}(H)=k$ and

Multi-robber damage conjecture. For all $s>2$, if

Asymptotic tightness conjecture. For any fixed positive integer $d$, we have

Let $G$ be a graph or digraph inducing a connectivity system, and consider an elimination game with an invisible robber in which cop positions satisfy the relevant connectivity bou…

Unbounded attacking-cop-number gap conjecture. For every non-negative integer $k$, there exists a graph $H$ such that

Planar attacking cop number conjecture. For every planar graph $G$,

A metric space $(X,d)$ is homogeneous if for every $x,y\in X$ there is an isometry $i:X\to X$ with $i(x)=y$. Let $X$ be a game space, namely a compact geodesic metric space equippe…

Let an MPSE problem have pursuers and evaders at positions at time $t$, with $0\leq t<t_c$. Let $\mathcal{B}^t$ be the Apollonius boundary and let $\mathcal{A}_i$ denote the Apollo…

Let $G$ be a connected graph of order $n$. In the Cat and Mouse localisation game, the cat receives only the information specified by the game rules and seeks to determine a vertex…

Logarithmic localization conjecture. The cat can localize the mouse up to distance $O(\log n)$ on $G$.