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

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.