$K_{5,5}$ is fully reconstructible in $\mathbb{C}^3$

TL;DR

This paper proves that the complete bipartite graph $K_{5,5}$ can be uniquely reconstructed from its 3-dimensional measurement variety in complex space, filling a key gap in the understanding of graph reconstructibility.

Contribution

The paper establishes that $K_{5,5}$ is fully reconstructible in $ ext{C}^3$, advancing the theory of graph reconstruction from measurement varieties.

Findings

$K_{5,5}$ is fully reconstructible in $ ext{C}^3$

Fills the gap in reconstructibility theory for $d=3$

Advances understanding of measurement variety-based graph reconstruction

Abstract

A graph $G$ is fully reconstructible in $\mathbb{C}^d$ if the graph is determined from its $d$-dimensional measurement variety. The full reconstructibility problem has been solved for $d=1$ and $d=2$. For $d=3$, some necessary and some sufficient conditions are known and $K_{5,5}$ falls squarely within the gap in the theory. In this paper, we show that $K_{5,5}$ is fully reconstructible in $\mathbb{C}^3$.