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We prove the existence of an upper bound on the asymptotic dimension of treeamalgamations of locally finite quasi-transitive connected graphs. Thisgeneralises a result of Dranishnikov for free products with amalgamation and aresult of Tselekidis for HNN-extensions of groups to tree amalgamations ofgraphs. As a corollary, we obtain an upper bound on the asymptotic dimension ofa multi-ended quasi-transitive locally finite graph based on any of theirfactorisations.

Asymptotic dimension of multi-ended quasi-transitive graphs