Sets admitting connection by graphs of finite length

The aim of this paper is the description and study of the properties of the subsets $M$ of a metric space $\mathbb X$ that can be connected by a graph of finite length. We obtain a criterion describing these sets, find several geometric properties of them (in the case $\mathbb X=\mathbb R^n$), and derive a formula for calculating the length of a minimal spanning tree on $M\subset\mathbb X$ as the integral of a certain function constructed by $M$.