MSTs are emblematic example of topological structures commonly used in communication networks. In a graph *G* = (*V*, *E*), an *MST* is a connected subset of the graph that contains all the vertices and minimizes the overall sum of its edge lengths (or more generally their *weight*). A well-known centralized algorithm to build such a structure is the Kruskal's algorithm, which mainly works as follows. All edges of *E* are sorted in the increasing weight order, and a new empty set *E*′ is created. For each edge *e* in the ordered set *E*, *e* is added to *E*′ if it does not create a loop in *G*′ = (*V*, *E*′). The resulting graph *G*′ is an MST of *G*.

The Figure 7.1 gives an example of *MST* that has been built over a random UDG. As usual, in wireless networks, the *length* of the edges was used as their weight. Note that if several edges have the same length, then several equivalent *MSTs* may exist, but this nondeterminism can be easily broken based on additional parameters (such as a comparison between nodes IDs).

An interesting fact about the MST is that its longest edge also corresponds to the optimal common transmission radius (i.e., the minimal radius such that network connectivity is preserved). This follows directly from Kruskal's algorithm, this edge being the last added edge in *E*′.

Considering a few related results, ...

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