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.
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Acknowledgements
The results of this paper were born in an MSc thesis. The author thanks Miklós Maróti, his supervisor, for his support.
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In the beginning, this research was supported by TÁ MOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program—Elaborating and operating an inland student and researcher personal support system”. This project was subsidized by the European Union and co-financed by the European Social Fund. Later, the research was supported by the Hungarian National Foundation for Scientific Research grant no. K115518.
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Kunos, Á. Definability in the Embeddability Ordering of Finite Directed Graphs, II. Order 36, 291–311 (2019). https://doi.org/10.1007/s11083-018-9467-2
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DOI: https://doi.org/10.1007/s11083-018-9467-2