A pseudo-inverse of a line graph
arXiv:2508.09412v1 Announce Type: new
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 ErdH{o}s-R’enyi graphs show that our theoretical results work in practice.
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 ErdH{o}s-R’enyi graphs show that our theoretical results work in practice.
Sevvandi Kandanaarachchi, Philip Kilby, Cheng Soon Ong
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