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

TL;DR

This paper develops a combinatorial framework to encode and analyze 2-connected subgraph patterns in polygon dissections and outerplanar graphs, deriving algebraic generating functions and asymptotic enumeration results.

Contribution

It introduces an explicit combinatorial decomposition that proves the algebraicity of pattern counting generating functions and enables pattern avoidance enumeration.

Findings

Generating functions are algebraic for counting pattern occurrences.

Derived normal limit laws for pattern distributions.

Performed asymptotic enumeration for pattern-avoiding classes.

Abstract

Let $\Delta =\{ \delta_1,\delta_2,...,\delta_m \} $ be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function $D_{\Delta }(z,u_1,u_2,...,u_m),$ which counts polygon dissections and marks subgraph copies of $\delta_i$ with the variable $u_i$. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on $\Delta $. The decomposition also gives a defining system for $D_{\Delta }(z,\mathbf{0})$, which encodes polygon dissections that restrict 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 directly transferred in the case of labelled outerplanar graphs. We give examples and compute the relevant constants when the patterns are small cycles or dissections.