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

Vasiliki Velona
2018 arXiv   pre-print
Let Δ ={δ_1,δ_2,...,δ_m } be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function D_Δ(z,u_1,u_2,...,u_m), which counts polygon dissections and marks subgraph copies of δ_i with the variable u_i. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on Δ . The decomposition also gives a defining system for D_Δ(z,0), which encodes polygon dissections that restrict these patterns as subgraphs. In
more » ... s 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.
arXiv:1802.03719v2 fatcat:u6pxjxmz4zcjlg57gdxzmr65ha