A Feasible Algorithm for Solubility of Transitive Graphs

Abstract

Transitive, weighted graphs, with the weights coming from an Abelian group and satisfying a transitivity property, were studied in [1] and were referred to as the transitive systems. Of central importance is the question of completing a given system to a transitive system where every ordered pair of the vertices can be assigned a weight. It was proven in [1] that, in the finite case, the question can be reduced to the situation when the Abelian group is a finite cyclic group. Nevertheless, in practical applications, the most common assignments of the weights are coming from the groups of numbers. For this reason, we consider the case when the underlying Abelian group G is torsion-free. We show that within the class of all torsion-free Abelian groups, the underlying transitive and unweighted graph predetermines the possibility of completion of a given transitive system. Such a graph is G-soluble. The question of G-solubility of a graph is equivalent to its ℝ-solubility. This, in turn, is equivalent to solvability of a certain system of linear equations in ℝ. As a result, G-solubility of a transitive graph can be determined by means of a feasible algorithm.