Vol. 9, No. 3, 2020

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First-order definitions of subgraph isomorphism through the adjacency and order relations

Oleg Grigoryan, Mikhail Makarov and Maksim Zhukovskii

Vol. 9 (2020), No. 3, 293–302

We study first-order definitions of graph properties over the vocabulary consisting of the adjacency and order relations. We compare logical complexities of subgraph isomorphism in terms of the minimum quantifier depth in two settings: with and without the order relation. We prove that, for pattern-trees, it is at least (roughly) two times smaller in the former case. We find the minimum quantifier depths of <-sentences defining subgraph isomorphism for all pattern graphs with at most 4 vertices.

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first-order logic, subgraph isomorphism, order, logical complexity, quantifier depth
Mathematical Subject Classification 2010
Primary: 03C13, 68Q19
Received: 3 December 2019
Revised: 6 February 2020
Accepted: 21 February 2020
Published: 15 October 2020
Oleg Grigoryan
Higher School of Economics
Mikhail Makarov
Moscow Institute of Physics and Technology
Maksim Zhukovskii
Laboratory of Advanced Combinatorics and Network Applications
Moscow Institute of Physics and Technology