Rigidity and reconstruction for graphs

TL;DR

This paper introduces a measure theoretic approach to graph rigidity, linking boundary measures of a 2b-regular tree to the graph's structure, and establishes equivalence between edge-reconstruction and closed walk lengths.

Contribution

It develops a novel measure theoretic framework for graph rigidity and proves the equivalence of edge-reconstruction with closed walk length reconstruction.

Findings

Measure theoretic rigidity for graphs with Betti number > 1

Explicit connection between boundary measures and graph structure

Edge-reconstruction is equivalent to closed walk length reconstruction

Abstract

We present measure theoretic rigidity for graphs of first Betti number b>1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths".