The Descriptive Complexity of Subgraph Isomorphism in the Absence of Order

Let C be a class of graphs and π be a graph parameter. Let Φ be a formula in the first-order language containing only the adjacency and the equality relations. We say that Φ defines C on connected graphs with sufficiently large π if there is a constant k such that, for every connected graph G with π(G) ≥ k, Φ is true on G exactly when G belongs to C. For a fixed connected graph F , let S(F ) denote the class of all graphs containing F as a subgraph. Let Dπ(F ) denote the minimum quantifier depth of a formula Φ defining S(F ) on connected graphs with sufficiently large π. We have Dv(F ) ≥ Dtw (F ) ≥ Dκ(F ), where v(G) denotes the number of vertices in a graph G, tw(G) is the treewidth of G, and κ(G) is the connectivity of G. For these three parameters we obtain the following results. • There are graphs F such that Dv(F ) is strictly smaller than the number n of vertices in F . In particular, Dv(Pn) = n − 1 for the path graphs on n ≥ 4 vertices. Moreover, there are some trees F such that Dv(F ) ≤ n− 3. • On the other hand, Dv(F ) = Dtw (F ) = n if F has no vertex of degree 1. In general, Dv(F ) > n/2 unless F = P2 or P3. • Dtw (F ) ≥ tw(F ) for every F . Over trees F with n vertices, the values of Dtw (F ) occupy the almost full spectrum {1, 5, . . . , n}. The minimum value Dtw (F ) = 1 is attained if F is a subtree of a subdivided 3-star K1,3. The maximum Dtw (K1,n−1) = n is attained for the star graphs on n ≥ 5 vertices. • Dκ(F ) ≥ m n +2 whenever the number m of edges in F is larger than the number n of vertices. Over graphs F with n vertices, the values of Dκ(F ) occupy the almost full spectrum {1, 3, . . . , n}. The minimum value Dκ(F ) = 1 is attained for any tree F . The maximum Dκ(Kn) = n is attained for the complete graphs on n ≥ 3 vertices.