Triangulating planar graphs while keeping the pathwidth small

TL;DR

This paper presents methods to triangulate planar graphs and outer-planar graphs while controlling the increase in pathwidth, improving previous bounds and maintaining structural properties.

Contribution

It introduces new techniques to triangulate planar graphs with minimal pathwidth increase and improves bounds for outer-planar graphs.

Findings

Triangulating planar graphs increases pathwidth by a factor of at most 8 or 16.

Outer-planar graphs can be made maximal with at most 4 times the original pathwidth plus 4.

New bounds improve upon previous results for both graph classes.

Abstract

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth $k$, then we can triangulate it so that the resulting graph has pathwidth $O(k)$ (where the factors are 1, 8 and 16 for 3-connected, 2-connected and arbitrary graphs). With similar techniques, we also show that any outer-planar graph of pathwidth $k$ can be turned into a maximal outer-planar graph of pathwidth at most $4k+4$. The previously best known result here was $16k+15$.