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On Categoricity Spectra for Locally Finite Graphs

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Under study is the algorithmic complexity of isomorphisms between computable copies of locally finite graphs \( G \) (undirected graphs whose every vertex has finite degree). We obtain the following results: If \( G \) has only finitely many components then \( G \) is \( {\mathbf{d}} \)-computably categorical for every Turing degree \( {\mathbf{d}} \) from the class \( PA({\mathbf{0}}^{\prime}) \). If \( G \) has infinitely many components then \( G \) is \( {\mathbf{0}}^{\prime\prime} \)-computably categorical. We exhibit a series of examples showing that the obtained bounds are sharp.

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The authors were supported by the Russian Foundation for Basic Research (Grant 20–31–70006).

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Correspondence to N. A. Bazhenov.

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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 5, pp. 983–994.

To Sergey Savost’yanovich Goncharov on the occasion of his 70th birthday.

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Bazhenov, N.A., Marchuk, M.I. On Categoricity Spectra for Locally Finite Graphs. Sib Math J 62, 796–804 (2021).

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