A pseudo-inverse of a line graph

TL;DR

This paper introduces a method to approximate the inverse of the line graph transformation by editing minimal edges in perturbed line graphs, supported by theoretical proofs and empirical validation.

Contribution

It proposes a linear integer programming approach to recover root graphs from perturbed line graphs, establishing theoretical properties and demonstrating practical effectiveness.

Findings

The pseudo-inverse operation is well-behaved under spectral norm.

Empirical results on Erdős-Rényi graphs validate the theoretical insights.

Abstract

Line graphs are an alternative representation of graphs where each vertex of the original (root) graph becomes an edge. However not all graphs have a corresponding root graph, hence the transformation from graphs to line graphs is not invertible. We investigate the case when there is a small perturbation in the space of line graphs, and try to recover the corresponding root graph, essentially defining the inverse of the line graph operation. We propose a linear integer program that edits the smallest number of edges in the line graph, that allow a root graph to be found. We use the spectral norm to theoretically prove that such a pseudo-inverse operation is well behaved. Illustrative empirical experiments on Erd\H{o}s-R\'enyi graphs show that our theoretical results work in practice.