On graphs which have locally complete 2-edge-colourings and their relationship to proper circular-arc graphs

TL;DR

This paper studies graphs with locally complete 2-edge-colourings, providing polynomial recognition algorithms, forbidden substructure characterizations, and connections to proper circular-arc graphs, advancing understanding of their structure and properties.

Contribution

It introduces polynomial algorithms for recognizing such graphs and characterizes them via forbidden substructures, linking to proper circular-arc graphs.

Findings

Recognition of these graphs is polynomial-time solvable.

Forbidden substructure characterization analogous to Gallai's for cocomparability graphs.

Characterization of proper circular-arc graphs with locally complete 2-edge-colourings.

Abstract

A 2-edge-coloured graph $G$ is called {\bf locally complete} if for each vertex $v$, the vertices adjacent to $v$ through edges of the same colour induce a complete subgraph in $G$. Locally complete 2-edge-coloured graphs have nice properties and there exists a polynomial algorithm to decide whether such a graph has an alternating hamiltonian cycle, where alternating means that the colour of two consecutive edges on the cycle are different. In this paper we show that graphs having locally complete 2-edge-colourings can be recognized in polynomial time. We give a forbidden substructure characterization for this class of graphs analogous to Gallai's characterization for cocomparability graphs. Finally, we characterize proper interval graphs and proper circular-arc graphs which have locally complete 2-edge-colourings by forbidden subgraphs.