A study of link graphs

TL;DR

This paper introduces link graphs, a new family of graphs generalizing line and path graphs, with characterizations, recognition algorithms, and properties related to chromatic number, minors, and structure.

Contribution

It defines link graphs, provides characterizations and recognition algorithms, and explores their structural properties and implications for graph theory.

Findings

Link graphs generalize line and path graphs.

Recognition problem for link graphs is in NP.

Structural properties of link graphs include bounds on chromatic and Hadwiger numbers.

Abstract

Graph theory is a branch of mathematics in which pair-wise relations between objects are studied. My PhD thesis, supervised by David R. Wood, introduces and investigates a new family of graphs, called link graphs, that generalises the notions of line graphs and path graphs. An s-link is a walk of length s such that consecutive edges are different. The s-link graph of a given graph G is the graph with vertices the s-links of G, and two vertices are adjacent if their corresponding s-links form an (s + 1)-link; Or equivalently, one corresponding s-link can be shunted to the other in one step. For example, the 1-link graph of G is the line graph of G. We give a characterisation for link graphs, which leads to algorithmic solutions to their recognition and determination problems, and implies that the recognition problem belongs to NP. Moreover, based on a recursive structure of linkgraphs, we obtain results about the chromatic number, Hadwiger number and isomorphism group of link graphs. We also obtain results about the uniqueness, tree-decomposition and better-quasi-ordering of the original graphs of a given link graph.