Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs

Let Δ={δ1,δ2,…,δm} be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function DΔ(z,u1,u2,…,um), which counts polygon dissections and marks subgraph copies of δi with the variable ui. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on Δ. The decomposition also gives a defining system for DΔ(z,0), which encodes polygon dissections that avoid these patterns as subgraphs. In this way, we are able to extract normal limit laws for the patterns when they are encoded, and perform asymptotic enumeration of the resulting classes when they are avoided. The results can be transferred to the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.