Abstract
A structure is called ultrahomogeneous if every isomorphism between finitely generated substructures of the structure extends to an automorphism of the structure. Recently, Cameron and Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In this paper we classify finite homomorphism-homogeneous oriented graphs with loops allowed which are uniform (that is, either all vertices have a loop or no vertex has a loop) or disconnected.
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The author would like to thank the three anonymous reviewers for many useful suggestions that significantly improved the clarity of the presentation.
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Supported by the Grant No. 174019 of the Ministry of Education and Science of the Republic of Serbia.
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Mašulović, D. Towards the Characterization of Finite Homomorphism-Homogeneous Oriented Graphs with Loops. Graphs and Combinatorics 31, 1613–1628 (2015). https://doi.org/10.1007/s00373-014-1435-z
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DOI: https://doi.org/10.1007/s00373-014-1435-z