Realization of distance matrices by graphs of genus 1

Abstract

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.