# Spanning Trees and Arborescences

• Bernhard Korte
• Jens Vygen
Chapter
Part of the Algorithms and Combinatorics book series (AC, volume 21)

## Abstract

Consider a telephone company that wants to rent a subset from an existing set of cables, each of which connects two cities. The rented cables should suffice to connect all cities and they should be as cheap as possible. It is natural to model the network by a graph: the vertices are the cities and the edges correspond to the cables. By Theorem  the minimal connected spanning subgraphs of a given graph are its spanning trees. So we look for a spanning tree of minimum weight, where we say that a subgraph T of a graph G with weights $$c: E(G) \rightarrow \mathbb{R}$$ has weight c(E(T)) = eE(T) c(e). We shall refer to c(e) also as the cost of e.

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