Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A pseudo-inverse of a line graph

Sevvandi Kandanaarachchi, Philip Kilby, Cheng Soon Ong

TL;DR

This work tackles the problem of inverting the line-graph operator when the observed graph is a perturbation of a line graph. It introduces a pseudo-inverse $L^{\dagger}$ that yields a root-like graph $\hat{G}$ by minimally editing the perturbed line graph $\widetilde{H}$ so that $\hat{H} = L(\hat{G})$ is a line graph, and provides an ILP formulation to compute these edits. A spectral-radius framework establishes boundedness and stability of the pseudo-inverse under single-edge perturbations, linking the root and line-graph spaces via $A(L(G)) = B'B - 2I$ and related norm bounds. Empirical results on Erdős–Rényi graphs and population-genetics data (haplotype population size estimation) demonstrate practical effectiveness and potential applications of the pseudo-inverse in graph recovery and inference.

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 Erdős-Rényi graphs show that our theoretical results work in practice.

A pseudo-inverse of a line graph

TL;DR

This work tackles the problem of inverting the line-graph operator when the observed graph is a perturbation of a line graph. It introduces a pseudo-inverse that yields a root-like graph by minimally editing the perturbed line graph so that is a line graph, and provides an ILP formulation to compute these edits. A spectral-radius framework establishes boundedness and stability of the pseudo-inverse under single-edge perturbations, linking the root and line-graph spaces via and related norm bounds. Empirical results on Erdős–Rényi graphs and population-genetics data (haplotype population size estimation) demonstrate practical effectiveness and potential applications of the pseudo-inverse in graph recovery and inference.

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 Erdős-Rényi graphs show that our theoretical results work in practice.

Paper Structure

This paper contains 30 sections, 43 theorems, 69 equations, 23 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Introducing a pseudo-inverse of a line graph
  4. The different cases
  5. Spectral radius bounds between $G$ and $H$ spaces
  6. Sensitivity to "small" perturbations
  7. Estimating a pseudo-inverse line graph
  8. Experiments
  9. Synthetic examples
  10. Estimating haplotype population size
  11. Conclusion
  12. Proof of Theorem \ref{['thm:DifferentCases']}
  13. Case I: edge addition in $H$ space
  14. Case III (a): edge addition and deletion in $G$ space
  15. Case III (b): vertex merging and splitting in $G$ space
  16. ...and 15 more sections

Key Result

Theorem 1

Let $G$ and $G'$ be connected graphs with isomorphic line graphs. Then $G$ and $G'$ are isomorphic unless one is $K_3$ and the other is $K_{1,3}$.

Figures (23)

  • Figure 1: Setting and notation. Given a graph $G$, we have the corresponding line graph $H:=L(G)$. $\widetilde{H}$ is a distorted version of $H$, which may not be a line graph. $\hat{H}$ is a closest line graph to $\widetilde{H}$, and $\hat{G}$ is a pseudo-inverse of $\widetilde{H}$.
  • Figure 2: A graph on the left and its line graph on the right.
  • Figure 3: The nine line-forbidden graphs as illustrated in beineke2021line
  • Figure 4: Line graphs $H_1$ and $H_2$, and their inverse line graphs $G_1$ and $G_2$ in the triangle closing scenario.
  • Figure 5: Graph $G_1$ on left with edges $a$ and $b$ not sharing a vertex and graph $G_2$ on the right with edges $a$ and $b$ sharing a vertex. Possible edges shown in dashed lines.
  • ...and 18 more figures

Theorems & Definitions (86)

  • Definition 1
  • Theorem 1: Whitney (1932), Harary (1969)
  • Theorem 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 76 more