A New Characterisation of Total Graphs

TL;DR

This paper introduces a new characterization of total graphs of simple graphs, providing improved algorithms for inverse total graph computation and insights into their local structure, especially for complete graphs.

Contribution

It offers a novel characterization of total graphs and develops efficient algorithms for their inverse, advancing previous BFS-based methods with structural insights.

Findings

New characterization of total graphs

Efficient algorithms for inverse total graphs

Partitioning of total graph vertices into original and line graphs

Abstract

Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a vertex colouring problem of the transformed graph. In this paper, we obtain a new characterisation of total graphs of simple graphs. We also design algorithms to compute the inverse total graphs when the input graph is a total graph. These results improve over the work of Behzad, by using novel observations on the properties of the local structure in the neighbourhood of each vertex. The earlier algorithm was based on bfs and distances. Our theorems result in partitioning the vertex set of the total graph into the original graph and the line graph efficiently. We obtain constructive results for special classes, most notably for total graph of complete graphs.