Imposing edges in Minimum Spanning Tree

TL;DR

This paper analyzes the impact of imposing additional edges on a minimum spanning tree, establishing a lower bound on the extra costs based on replacement costs of these edges.

Contribution

It proves that the sum of replacement costs of imposed edges provides a lower bound on the total cost of a spanning tree containing those edges.

Findings

Sum of replacement costs is a lower bound for additional costs.

Imposing edges affects the minimum spanning tree cost.

Theoretical bounds on cost increase due to imposed edges.

Abstract

We are interested in the consequences of imposing edges in $T$ a minimum spanning tree. We prove that the sum of the replacement costs in $T$ of the imposed edges is a lower bounds of the additional costs. More precisely if r-cost$(T,e)$ is the replacement cost of the edge $e$, we prove that if we impose a set $I$ of nontree edges of $T$ then $\sum_{e \in I} $ r-cost$(T,e) \leq$ cost$(T_{e \in I})$, where $I$ is the set of imposed edges and $T_{e \in I}$ a minimum spanning tree containing all the edges of $I$.