Polynomial-time solvability of the Misère Domination Game on sparse graph classes

Determine whether the Misère Domination Game is polynomial-time solvable on paths and cycles, and whether it is polynomial-time solvable on trees, under the convention that the last player to move loses.

Background

For the normal-play Domination Game, the paper derives exact results for paths and cycles and a polynomial-time solution for disjoint unions of paths and cycles. The authors note that the Sprague–Grundy theory is more complicated for misère games and leave unresolved the corresponding algorithmic questions for paths, cycles, and trees.