The Descriptive Complexity of Subgraph Isomorphism in the Absence of Order

Let $C$ be a class of graphs and $\pi$ be a graph parameter. Let $\Phi$ be a formula in the first-order language containing only the adjacency and the equality relations. We say that $\Phi$ \emph{defines $C$ on connected graphs with sufficiently large $\pi$} if there is a constant $k$ such that, for every connected graph $G$ with $\pi(G)\ge k$, $\Phi$ 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 $D_\pi(F)$ denote the minimum quantifier depth of a formula $\Phi$ defining $S(F)$ on connected graphs with sufficiently large $\pi$. We have $D_v(F)\ge D_{tw}(F)\ge D_\kappa(F)$, where $v(G)$ denotes the number of vertices in a graph $G$, $tw(G)$ is the treewidth of $G$, and $\kappa(G)$ is the connectivity of~$G$. We obtain the following results. - There are graphs $F$ such that $D_v(F)$ is strictly smaller than the number $n$ of vertices in $F$. In particular, $D_v(P_n)=n-1$ for the path graphs on $n\ge4$ vertices. Moreover, there are some trees $F$ such that $D_v(F)\le n-3$. - On the other hand, $D_v(F)=D_{tw}(F)=n$ if $F$ has no vertex of degree 1. In general, $D_v(F)>n/2$ unless $F=P_2$ or $P_3$. - $D_{tw}(F)\ge tw(F)$ for every $F$. Over trees $F$ with $n$ vertices, the values of $D_{tw}(F)$ occupy the almost full spectrum $\{1,5,\ldots,n\}$. The minimum value $D_{tw}(F)=1$ is attained if $F$ is a subtree of a subdivided 3-star $K_{1,3}$. The maximum $D_{tw}(K_{1,n-1})=n$ is attained for the star graphs on $n\ge5$ vertices. - $D_\kappa(F)\ge\frac mn+2$ 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 $D_\kappa(F)$ occupy the almost full spectrum $\{1,3,\ldots,n\}$.