On triangulating k-outerplanar graphs

TL;DR

This paper investigates the triangulation of k-outerplanar graphs, demonstrating that while they cannot always be triangulated to remain k-outerplanar, they can be triangulated to become (k+1)-outerplanar, balancing planarity and triangulation.

Contribution

It provides a new understanding of the limitations and possibilities in triangulating k-outerplanar graphs while controlling their outerplanarity.

Findings

Not all k-outerplanar graphs can be triangulated to stay k-outerplanar.

They can be triangulated to become (k+1)-outerplanar.

The results inform graph drawing and planar graph algorithms.

Abstract

A k-outerplanar graph is a graph that can be drawn in the plane without crossing such that after k-fold removal of the vertices on the outer-face there are no vertices left. In this paper, we study how to triangulate a k-outerplanar graph while keeping its outerplanarity small. Specifically, we show that not all k-outerplanar graphs can be triangulated so that the result is k-outerplanar, but they can be triangulated so that the result is (k+1)-outerplanar.