Undirected edge geography games on stacked prism graphs

The undirected edge geography is a two-player combinatorial game on an undirected graph. The players start at the root vertex and alternately move the root along an incident edge to its other endpoint and then delete that edge. The first player who has no remaining move is the loser. For positive integers $m$ and $n$ where $m\ge 3$, the stacked prism graph is $SP(m,n)=C_m\mathrm{\square }{P}_n$. In this paper, we completely determine the winner of the game on $SP(2m,n)$ for $m\ge 2$ and $SP(m,2)$ for $m\ge 3$, and provide a winning strategy for the winner.