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<p>The modern theory of homogeneous structures begins with the work of Roland Fraïssé. The theory developed in the last seventy years is placed in the border area between combinatorics, model theory, algebra, and analysis. We turn our attention to its combinatorial pillar, namely, the work on the classification of structures for given homogeneity types, and focus onto the homomorphism homogeneous ones, introduced in 2006 by Cameron and Nešetřil. An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs. Among these we identify those that are polymorphism homogeneous. Here an oriented graph is called <i>polymorphism homogeneous</i> if each of its finite powers is homomorphism homogeneous.</p>

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The Classification of Homomorphism Homogeneous Oriented Graphs

  • 期刊论文

  • 发表日期：

    2026年3月20日

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  • Bojana Pavlica,
  • Christian Pech,
  • Maja Pech

摘要

The modern theory of homogeneous structures begins with the work of Roland Fraïssé. The theory developed in the last seventy years is placed in the border area between combinatorics, model theory, algebra, and analysis. We turn our attention to its combinatorial pillar, namely, the work on the classification of structures for given homogeneity types, and focus onto the homomorphism homogeneous ones, introduced in 2006 by Cameron and Nešetřil. An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs. Among these we identify those that are polymorphism homogeneous. Here an oriented graph is called polymorphism homogeneous if each of its finite powers is homomorphism homogeneous.