Minimum spanning tree formulation

Question

Can any expert explain the reasoning behind the constraint in the following formulation of the minimum spanning tree?

To formulate the minimum-cost spanning tree (MST) problem as an LP, we associate a variable $x_e$ with every edge $e \in E$. Each spanning tree $T$ corresponds to its incidence vector $x^T$, which is defined by $x^T_e = 1$ if $T$ contains $e$ and $x^T_e = 0$ otherwise. Let $\Pi$ denote the set of all partitions of the vertex set $V$, and suppose that $\pi \in \Pi$. The rank $r(\pi)$ of $\pi$ is the number of parts of $\pi$. Let $E_\pi$ denote the set of edges whose ends lie in different parts of $\pi$. Consider the following LP: $$ \begin{align*} \min &\sum_{e \in E} c_ex_e \\ \text{s.t. } &\sum_{e \in E_\pi} x_e \geq r(\pi) - 1 \quad \forall \pi \in \Pi, \\ & x \geq 0. \\ \end{align*} $$

Answer 1

The constraint expresses the following fact:

Let $T$ be a spanning tree, and suppose that we partition the vertex set into $r$ parts. There are exactly $r-1$ edges of $T$ which connect different parts.

For example, if $T$ is a tree and $C,\overline{C}$ is a cut, then exactly one edge of $T$ crosses the cut, that is, connects $C$ and $\overline{C}$.