Decomposing Triangulations into 4-Connected Components

TL;DR

This paper presents a simple, easy-to-implement linear-time algorithm based on depth-first search for decomposing triangulated graphs into 4-connected components and determining their nesting structure, improving practical implementation over previous methods.

Contribution

It introduces a new, straightforward linear-time algorithm for decomposing triangulated graphs into 4-connected components, addressing implementation difficulties of prior approaches.

Findings

The algorithm runs in linear time.

It effectively decomposes triangulated graphs into 4-connected components.

It accurately computes the nesting structure of components.

Abstract

A connected graph is 4-connected if it contains at least five vertices and removing any three of them does not disconnect it. A frequent preprocessing step in graph drawing is to decompose a plane graph into its 4-connected components and to determine their nesting structure. A linear-time algorithm for this problem was already proposed by Kant. However, using common graph data structures, we found the subroutine dealing with triangulated graphs difficult to implement in such a way that it actually runs in linear time. As a drop-in replacement, we provide a different, easy-to-implement linear-time algorithm that decomposes a triangulated graph into its 4-connected components and computes the respective nesting structure. The algorithm is based on depth-first search.