Graphs measure interactions between objects such as friendship links on Twitter,
transactions between Bitcoin users, and the flow of energy in a food chain. While graphs …

We introduce an “intermediate branching number”(IBN) which captures the branching of
intermediate growth trees, similar in spirit to the well-studied branching number of …

We prove that any Cayley graph G with degree d polynomial growth does not satisfy {f (n)}-
containment for any f= o (nd− 2). This settles the asymptotic behaviour of the firefighter …

We study the fire-retaining problem on groups, a quasi-isometry invariant 1 introduced by
Martínez-Pedroza and Prytuła [8], related to the firefighter problem. We prove that any …

In the Firefighter problem, introduced by Hartnell in 1995, a fire spreads through a graph
while a player chooses which vertices to protect in order to contain it. In this paper, we focus …

The weak-cop number of a graph, introduced by Lee et al (2023), is a quasi-isometry
invariant of graphs and hence of finitely generated groups. While for any $ m\in\mathbb {Z} …

Given a non-decreasing function f: N→ N we define a single player game on (infinite)
connected graphs that we call fire retaining. If a graph G admits a winning strategy for any …

The containment game is a full information game for two players, initialised with a set of
occupied vertices in an infinite connected graph G. On the t-th turn, the first player, called …

We define two impartial games, the Relator Achievement Game REL and the Relator
Avoidance Game RAV. Given a finite group G and generating set S, both games begin with …

Given a non-decreasing function $ f\colon\mathbb {N}\to\mathbb {N} $ we define a single
player game on (infinite) connected graphs that we call fire retaining. If a graph $ G $ admits …