Whitney's Theorem for Line Graphs of Multi-Graphs

TL;DR

This paper extends Whitney's Theorem to line graphs of multi-graphs, showing most are uniquely determined by their line graphs and providing an algorithm to reconstruct such multi-graphs from a given line graph.

Contribution

It generalizes Whitney's Theorem to multi-graphs and introduces an algorithm for reconstructing multi-graphs from their line graphs.

Findings

Most multi-graphs are uniquely determined by their line graphs.

An algorithm is provided to identify if a graph is a line graph of a multi-graph.

The extension applies to both 1-line and ≥1-line graphs of multi-graphs.

Abstract

Whitney's Theorem states that every graph, different from $K_3$ or $K_{1,3}$, is uniquely determined by its line graph. A $1$-line graph of a multi-graph is the graph with as vertices the edges of the multi-graph, and two edges adjacent if and only if there is a unique vertex on both edges. The $\geq 1$-line graph of a multi-graph is the graph on the edges of the multi-graph, where two edges are adjacent if and only if there is at least one vertex on both edges. We extend Whitney's theorem to such line graphs of multi-graphs, and show that most multi-graphs are uniquely determined by their line graph. Moreover, we present an algorithm to determine for a given graph $\Gamma$, if possible, a multi-graph with $\Gamma$ as line graph.