The First Order Definability of Graphs with Separators via the Ehrenfeucht Game

We say that a first order formula $\mathrm{\Phi }$ defines a graph $G$ if $\mathrm{\Phi }$ is true on $G$ and false on every graph $G^{\mathrm{\prime }}$ non-isomorphic with $G$. Let $D(G)$ be the minimal quantifier rank of a such formula. We prove that, if $G$ is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then $D(G)=O(\log n)$, where $n$ denotes the order of $G$. This bound is optimal up to a constant factor. If $h$ is a constant, for connected graphs with no minor $K_h$ and degree $O(\sqrt{n}\mathrm{/}\log n)$, we prove the bound $D(G)=O(\sqrt{n})$. This result applies to planar graphs and, more generally, to graphs of bounded genus.