We say that a first order formula Ī¦ defines a graph G if Ī¦ is
true on G and false on every graph Gā² 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(logn), 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 Khā and degree O(nā/logn), we prove the bound
D(G)=O(nā). This result applies to planar graphs and, more generally, to
graphs of bounded genus.Comment: 17 page