We study the unbiased Maker-Breaker $K_{1,3}$-game, the unbiased claw game, played on the edge set of a general graph $G$. In this game, Maker and Breaker alternatingly claim edges of $G$, and Maker wins if his claimed edges contain a claw once no more edges are available. Graphs with maximum degree at least 5 are Maker's win. A graph $G$ is claw-connected if the hypergraph modeling the unbiased claw game played on $G$ is connected. We characterize Maker's win in the unbiased claw game on every claw-connected graph whose maximum degree is different from 3. Our main structural result is a complete characterization of Maker's win for claw-connected graphs with exactly one vertex of degree 4. The analysis is based on the fact that vertices of degree 4 act as forcing structures. Our characterization results in a linear-time algorithm for the unbiased claw game on claw-connected graphs with maximum degree different from 3. We also obtain a linear-time decomposition algorithm reducing the game on general graphs to a collection of pairwise edge-disjoint claw-connected subgraphs of maximum degree exactly 3, thereby isolating the only remaining case. This decomposition yields an $\mathcal{O}(|V(G)|^3)$-time algorithm for the unbiased claw game on general graphs, improving on the $\mathcal{O}(|V(G)|^7)$-time algorithm implied by the work of Galliot, Gravier and Sivignon (arXiv 2026). For a lower bound on the decision complexity, we prove that any agent deciding the game in the adjacency-array model makes at least $Ω(|V(G)|)$ queries in the worst case, indicating the optimality of our linear-time algorithm.

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