Counting Hexagonal Patches and Independent Sets in Circle Graphs

TL;DR

This paper presents the first polynomial-time algorithm for counting hexagonal patches with a given outer degree sequence, by reducing the problem to counting maximum independent sets in circle graphs, with applications in computational chemistry.

Contribution

It introduces a novel polynomial-time algorithm for counting specific plane graphs, linking it to circle graph independent set problems, and advances computational methods in chemistry-related graph enumeration.

Findings

First polynomial-time algorithm for counting hexagonal patches.

Reduction of the counting problem to maximum independent sets in circle graphs.

Efficient algorithm for counting independent sets in circle graphs.

Abstract

A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the study of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem.