Encoding and avoiding 2-connected patterns in polygon dissections and
outerplanar graphs
release_u6pxjxmz4zcjlg57gdxzmr65ha
by
Vasiliki Velona
2018
Abstract
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 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.
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