19 problems

Let $G$ be a connected chordal graph, meaning that it has no induced cycle of length greater than $3$, and let $\rho\ge 2$. A weak retract of $G$ is a graph obtained from $G$ by a…

Star-graph critical-speed conjecture. For every $k\geq 3$ and every choice of edge lengths $\bm{\ell}$,

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…

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…

Let $X$ be a simplicial complex homeomorphic to a compact manifold, endowed with its simplicial metric. Let $c(X)$ be its cop number and $c_0(X)$ its strong cop number. The propose…

In continuous pursuit–evasion games, the players have prescribed start points, the pursuer may capture the escaper by approaching within arbitrarily small positive distance, and th…

Consider the approximation algorithms described in Sections O(1) and pseudoPTAS for pursuit–escape games in polygonal domains. The 3D approximation conjecture. These approximation…

A pursuit–escape game has an escaper, a pursuer, motion-path domains with possibly different metrics or speeds, and winning conditions based on reaching an exit while maintaining d…

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 $G$.

Let $t_n$ denote the relevant capture-time quantity for the toroidal grid of side length $n$, and let $z(T_n)$ denote the minimum number of zombies needed to catch the survivor on…

Let $G$ be a finite, simple graph. Write $\xi(G)$ for its containability number, $c(G)$ for its cop number, and let $\Delta(G)$ denote its maximum degree. Komarov–Mackey's containa…

Locatable-graph colourability conjecture. Every locatable graph is $3$-colourable.

Let $G$ be a finite graph, and let $G^{1/m}$ be the graph obtained by replacing every edge of $G$ by a path of length $m$ through new vertices. A graph is locatable if the cop has…