Stacking numbers for eternal domination

Abstract

In this paper we study a new graph parameter, the stacking number. Defined in relation to the eternal domination game, we show that there are highly connected graphs for which it is beneficial to allow multiple guards to occupy a vertex, answering an open question of Finbow et al. In fact, we show that for any sequence \( (s_i) \), allowing \( s_j \) guards to occupy a vertex can save arbitrarily many guards in comparison to allowing fewer than this on a vertex. We also show that the stacking number is \( 1 \) for all trees.