On the hardness of recognizing triangular line graphs

TL;DR

This paper proves that recognizing triangular line graphs is NP-complete, establishing its computational complexity and contributing to understanding their structural properties.

Contribution

The paper demonstrates that recognizing triangular line graphs is NP-complete through a reduction from 3-SAT, resolving an open question about their computational complexity.

Findings

Recognition of triangular line graphs is NP-complete.

Triangular line graphs are computationally hard to identify.

The complexity status was previously unknown.

Abstract

Given a graph G, its triangular line graph is the graph T(G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and link graphs. While closely related to line graphs, triangular line graphs have been difficult to understand and characterize. Van Bang Le asked if recognizing triangular line graphs has an efficient algorithm or is computationally complex. We answer this question by proving that the complexity of recognizing triangular line graphs is NP-complete via a reduction from 3-SAT.