An exact-arithmetic algorithm for spanning tree modulus

TL;DR

This paper introduces an exact-arithmetic algorithm for spanning tree modulus, connecting it to graph vulnerability, and demonstrates its implementation and computational efficiency.

Contribution

It presents a novel exact-arithmetic algorithm for spanning tree modulus, leveraging Cunningham's vulnerability algorithm and integer arithmetic for practical application.

Findings

Algorithm efficiently computes spanning tree modulus

Connection established between spanning tree modulus and graph vulnerability

Demonstrated practical implementation and scaling results

Abstract

Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with spanning tree modulus is known to produce two hierarchical decompositions of arbitrary graphs, one based on strength and the other on arboricity. Here we introduce an exact-arithmetic algorithm for spanning tree modulus and the strength-based decomposition using Cunningham's algorithm for graph vulnerability. The algorithm exploits an interesting connection between spanning tree modulus and critical edge sets from the vulnerability problem. This paper introduces the new algorithm, describes a practical means for implementing it using integer arithmetic, and presents some examples and computational time scaling tests.