一个结构称为齐次的，如果该结构的每个子结构之间的同构都延伸到该结构的一个自同构。最近，P. J. Cameron和J. Nešetziil引入了一个放松版本的齐性：我们说一个结构是同态齐性的，如果该结构的每一个导生子结构之间的同态都延伸到该结构的一个自同态。本文考虑具有一个自反二元关系的有限同态-齐次关系系统。我们表明，对于很大一部分这样的关系系统（双向连接有向图;一个有向图是双向连接的，如果它的每个连接组件可以遍历的$\rightleftarrows$ -路径）的问题，决定系统是否是同态齐次是coNP-完全的。因此，对于这类关系系统，我们不能指望有一个包含目录的描述，因为通过目录我们可以理解一个有限的多项式可判定的结构类列表。另一方面，在双向不连通有向图的情况下，我们提出了完整的特征。我们的主要结果表明，如果一个有向图是双向不连通的，那么它是同态齐次的当且仅当它是有限同态齐次拟序，或者是同态齐次有向图的对合膨胀（本文后面介绍的一类特殊的有向图），或者是连通分支为$C_3^\circ$和$\1^\circ$的有向图的膨胀.
英文摘要：
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. Nešetřil 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 $\rightleftarrows$-paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems we cannot hope for a description involving a catalogue, where by a catalogue we understand a finite list of polynomially decidable classes of structures. 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 peculiar class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are $C_3^\circ$ and $\1^\circ$.
Finite irreflexive homomorphism-homogeneous binary relational systems
如果一个结构的有限子结构之间的每一个同构都延伸到该结构的一个自同构，则该结构被称为齐次的。最近，P. J. Cameron和J. Nešetziil引入了一个放松版本的齐性：我们说一个结构是同态齐性的，如果结构的有限子结构之间的每个同态都延伸到结构的自同态。本文刻画了具有一个非自反二元关系的有限同态-齐次关系系统。
