On the structure of graphs with path-width at most two

TL;DR

This paper characterizes the structure of graphs with path-width at most two by analyzing building blocks and their connections, providing a clearer understanding than previous extensive lists of excluded minors.

Contribution

It offers a structural characterization of graphs with path-width at most two, focusing on building blocks and their assembly, improving upon prior exhaustive lists.

Findings

Characterization of 2-connected graphs with path-width at most two

Characterization of 2-edge-connected graphs with path-width at most two

Outline of a complete structural characterization of all such graphs

Abstract

Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with path-width at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight to structural properties. We take a different route, and concentrate on the building blocks and how they are glued together. In this way, we get a characterization of 2-connected and 2-edge-connected graphs with path-width at most two. Along similar lines, we sketch the complete characterization of graphs with path-width at most two.