A tight bound on the collection of edges in MSTs of induced subgraphs

TL;DR

This paper establishes a tight upper bound on the total number of edges in all MSTs of induced subgraphs of a complete graph with distinct weights, confirming a conjecture and generalizing Mader's Theorem.

Contribution

It proves a conjecture by Goemans and Vondrak and extends Mader's Theorem by providing a precise bound on edges in MST collections of induced subgraphs.

Findings

The set of edges in all MSTs of induced subgraphs has at most nk - (k+1 choose 2) elements.

The result confirms the conjecture of Goemans and Vondrak.

The theorem generalizes Mader's Theorem on edge bounds in k-connected graphs.

Abstract

Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\in\{1,2,...,n-1\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$ vertices has at most $nk-\binom{k+1}{2}$ elements. This proves a conjecture of Goemans and Vondrak \cite{GV2005}. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal $k$-connected graph.