On finite reflexive homomorphism-homogeneous binary relational systems

TL;DR

This paper investigates finite reflexive binary relational systems that are homomorphism-homogeneous, revealing complexity results for bidirectionally connected systems and providing a full characterization for bidirectionally disconnected systems.

Contribution

It introduces a classification of finite homomorphism-homogeneous reflexive digraphs, including complexity results and a complete characterization for disconnected cases.

Findings

Deciding homomorphism-homogeneity in bidirectionally connected digraphs is coNP-complete.

Bidirectionally disconnected homomorphism-homogeneous digraphs are fully characterized.

The paper identifies specific classes of digraphs that are homomorphism-homogeneous.

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. Ne\v{s}et\v{r}il 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 $\1^\circ$.