On the logical complexity of convex polygon dissections

The logical depth of a graph $G$ is the minimum quantifier depth of a first order sentence defining $G$ up to isomorphism in the language of the adjacency and the equality relations. We consider the case that $G$ is a dissection of a convex polygon or, equivalently, a biconnected outerplanar graph. We bound the logical depth of a such $G$ from above by a function of combinatorial parameters of the dual tree of $G$.