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
Abstract

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.

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