# A local characterization for perfect plane near-triangulations

## TL;DR

This paper presents a local criterion to determine the perfectness of plane near-triangulated graphs, leading to an efficient $O(n^2)$ algorithm for checking their perfectness.

## Contribution

It introduces a novel local characterization for perfect plane near-triangulations and provides an efficient algorithm based on this criterion.

## Key findings

- A plane near-triangulated graph is perfect if and only if it lacks certain local configurations.
- The paper provides an $O(n^2)$ algorithm for testing perfectness in these graphs.
- The characterization simplifies the verification process for perfectness in plane near-triangulations.

## Abstract

We derive a local criterion for a plane near-triangulated graph to be perfect. It is shown that a plane near-triangulated graph is perfect if and only if it does not contain either a vertex, an edge or a triangle, the neighbourhood of which has an odd hole as its boundary. The characterization leads to an $O(n^2)$ algorithm for checking perfectness of plane near-triangulations.