is almost locally connected if \(B(G)\) is an independent set and for any \(x \in B(G)\), there is a vertex \(y\) in \(V(G) \setminus \{x\}\) such that \(N(x) \cup \{y\}\) induces a connected subgraph of \(G\), where \(B(G)\) denotes the set of vertices of \(G\) that are not locally connected. In this paper, we prove that an almost locally connected claw-free graph on at least \(4\) vertices is Hamilton-connected if and only if it is \(3\)-connected. This generalizes a result by Asratian that a locally connected claw-free graph on at least \(4\) vertices is Hamilton-connected if and only if it is \(3\)-connected [Journal of Graph Theory \(23 (1996) 191-201\)].
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