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arXiv · 0705.2439

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

Gregory B. Sorkin·Angelika Steger·Rico Zenklusen
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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.

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Gregory B. Sorkin, Angelika Steger, Rico Zenklusen. 2007-05-16. A tight bound on the collection of edges in MSTs of induced subgraphs. https://arxiv.org/abs/0705.2439

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