# PIMS-UVic Discrete Math Seminar: Ethan Williams

## Topic

Stacking Number for Eternal Domination

## Details

Eternal domination is a graph game where mobile guards are placed on vertices and then moved in order to defend against sequences of attacks. If only one guard is allowed to move in response to an attack, then it has been shown that there is no benefit to allowing multiple guards to occupy a vertex. It was conjectured that if all guards can move there would similarly be no benefit from multiple guards occupying a vertex. Finbow et al. showed this was untrue, and provided a construction demonstrating that there are graphs which need fewer guards when 2 guards are allowed on the same vertex. We extend these results to show that for any $k$ and $s$ there are $k$-connected graphs which need fewer guards when up to $s$ guards are allowed on any vertex.

This is joint work with Georgia Penner.

**Scientific, Seminar**

**November 23, 2023**

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