  Given a distance matrix $D$, we study the behavior of its compaction vector
and reduction matrix with respect to the problem of the realization of $D$ by a
weighted graph. To this end, we first give a general result on realization by
$n-$cycles and successively we mainly focus on graphs of genus 1, presenting an
algorithm which determines when a distance matrix is realizable by such a kind
of graph, and then, shows how to construct it.
