### Abstract

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Neetil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure. In this paper, we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectionally connected digraphs; a digraph is bidirectionally connected if each of its connected components can be traversed by ⇄-paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems there is no polynomially computable characterization (unless P=NP). On the other hand, in case of bidirectionally disconnected digraphs we present the full characterization. Our main result states that if a digraph is bidirectionally disconnected, then it is homomorphism-homogeneous if and only if it is either a finite homomorphism-homogeneous quasiorder, or an inflation of a homomorphism- homogeneous digraph with involution (a specific class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are C3° and 1^{°}.

Original language | English |
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Pages (from-to) | 2543-2555 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 21 |

DOIs | |

Publication status | Published - 6 Nov 2011 |

Externally published | Yes |

### Keywords

- Finite digraphs
- Homomorphism-homogeneous structures

## Cite this

*Discrete Mathematics*,

*311*(21), 2543-2555. https://doi.org/10.1016/j.disc.2011.07.032