Definability in the Embeddability Ordering of Finite Directed Graphs, II

Abstract

We deal with first-order definability in the embeddability ordering $(\mathscr{D}; \leq )$ of finite directed graphs. A directed graph $G\in \mathscr{D}$ is said to be embeddable into $G^{\prime } \in \mathscr{D}$ if there exists an injective graph homomorphism $\varphi \colon G \to G^{\prime }$. We describe the first-order definable relations of $(\mathscr{D}; \leq )$ using the first-order language of an enriched small category of digraphs. The description yields the main result of the author’s paper (Kunos, Order 32(1):117–133, 2015) as a corrolary and a lot more. For example, the set of weakly connected digraphs turns out to be first-order definable in $(\mathscr{D}; \leq )$. Moreover, if we allow the usage of a constant, a particular digraph A, in our first-order formulas, then the full second-order language of digraphs becomes available.