Minimal pentagulations of $n$-gons

TL;DR

This paper classifies minimal 3-connected pentagulations of n-gons for n between 3 and 12, using computational methods to identify the smallest such graphs with all faces pentagons except one.

Contribution

It provides a complete enumeration of minimal pentagulations for n-gons with 3 ≤ n ≤ 12, expanding understanding of these specialized planar graphs.

Findings

Minimal pentagulations for n=3 and n=4 contain 15 and 14 pentagons respectively.

All minimal pentagulations for 3 ≤ n ≤ 12 are explicitly determined.

The study employs computational generation of planar graphs using the plantri package.

Abstract

A planar graph $G$ is called a pentagulation of an $n$-gon ($n\geq$ is an integer) if all faces of $G$ are pentagons, except one, which is an $n$-gon. A $3$-connected pentagulation $G$ of an $n$-gon is called minimal if it has the smallest number of pentagons among all such $3$-connected pentagulations. It is known that minimal pentagulations of the $3$-gon and $4$-gon contain 15 and 14 pentagons, respectively. We determined all minimal pentagulations of $n$-gons for all $n$ such that $3\leq n\leq 12$ using computer calculations. The calculations employed the plantri package, which generates all planar triangulations for a given number of vertices. We also present several open questions on this topic.