Definability in the embeddability ordering of finite directed graphs, II

We deal with first-order definability in the embeddability ordering $( $\mathcal{D}$; \leq)$offinitedirectedgraphs\mathrm{.}Adirectedgraph$G\in $\mathcal{D}$$issaidtobeembeddableinto$G’ \in $\mathcal{D}$$ if there exists an injective graph homomorphism $\phi \text{\hspace{0.17em}⁣}:G\to G^{\mathrm{\prime }}$. We describe the first-order definable relations of $(\mathcal{D};\le )$ using the first-order language of an enriched small category of digraphs. The description yields the main result of one of the author’s papers as a corollary and a lot more. For example, the set of weakly connected digraphs turns out to be first-order definable in $(\mathcal{D};\le )$. 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.