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

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.