The first order definability of graphs with separators via the Ehrenfeucht game

We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph G 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 Kh and degree O( √ n/ log n), we prove the bound D(G) = O( √ n). This result applies to planar graphs and, more generally, to graphs of bounded genus. Our proof techniques are based on the characterization of the quantifier rank as the length of the Ehrenfeucht game on non-isomorphic graphs. We use the separator theorems to design a winning strategy for Spoiler in this game.