The classification of homomorphism homogeneous oriented graphs

TL;DR

The paper achieves a complete classification of countable homomorphism homogeneous oriented graphs without loops by reducing to their cores, which must be one of $I_1$, $C_3$, $(\mathbb{Q},<)$, $S(2)$, or $T^{\infty}$, and showing every HH graph is obtained from a core via a $\Gamma[f]$-style construction over a homogeneous tournament. It then separately classifies weakly connected and disconnected HH and PH oriented graphs, obtaining explicit lists: HH weakly connected graphs include acyclic strict posets, certain cyclic constructions $C_3[f]$, $S(2)[f]$, $T^{\infty}[f]$ with surjective $f$, and a few core-based families; PH weakly connected graphs reduce to $I_1$, $C_3$, $(\mathbb{Q},<)$, and extensions of the universal strict poset. In the disconnected case, HH graphs are unions of $k$ copies of a single homogeneous tournament, with base components drawn from $I_1$, $C_3$, $(\mathbb{Q},<)$, $S(2)$, or $T^{\infty}$, while PH admits only $k\cdot I_1$ and $k\cdot C_3$. The work also situates these results relative to Cherlin’s homogeneous oriented graphs and discusses the roles of ages and cores, highlighting both the finite scope of HH equivalence classes and the richness of age-structured families such as HH trees.

Abstract

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.

Figures (4)

• Figure 1: The posets $\mathbf{A}_n$
• Figure 2: From the word $w=0101$ to the tournament $T_w$
• Figure 3: From from a finite tournament $T<S(2)$ to the word $w_T=1011$
• Figure 4: From a finite graph $\Gamma<S(2)[I_2]$ to the word $w_\Gamma=(1,1)(0,1)(1,2)(1,1)$

Theorems & Definitions (59)

• Definition 2.1
• Proposition 2.2: PecPec16b
• proof
• Lemma 2.3
• proof
• Remark
• Corollary 2.4
• Proposition 2.5
• Lemma 2.6: Col20
• proof