Imposing edges in Minimum Spanning Tree

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$(T,e)\le$ 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 $I$.