An Upper Bound for the Isolation Game on Trees of Diameter at Most Six

In the isolation game, Dominator and Staller alternately select vertices of a graph. A move is legal if it dominates a vertex in a nontrivial component of the subgraph induced by the vertices not yet dominated. Dominator minimizes the number of moves, and Staller maximizes it. We prove that every tree T of order n ≥ 3 and diameter at most six satisfies ιg(T) ≤ ⌊(2n+1)/5⌋ ≤ 3n/7, where ιg denotes the Dominator-start game isolation number. The proof uses a vertex of eccentricity at most three: after this vertex is selected, the remaining nontrivial undominated components are stars grouped by their common dominated parent. A legal move removes one star or an entire group, which allows an elementary counting argument. Three small configurations require a different first move.