Multiset guards variant on trees

Investigate the complexity of Fast-Strategy when multiple guards per vertex are allowed (guards form a multiset), specifically determine whether polynomial-time algorithms exist on trees given that the naive twin-expansion reduction does not preserve the class.

Background

The paper primarily considers at most one guard per vertex. It notes a generic reduction from multisets by expanding vertices into true twins, which preserves cographs (maintaining the polynomial algorithm) but not trees.

This raises an open issue: whether one can develop direct algorithms or complexity results for trees with multiset guards, as existing reductions and methods do not straightforwardly apply.