Let G a weighted connexe graph. In this paper, we are interesting by covering the vertices of G by connexe disjoints unicyclic components with minimum total cost. We propose two polynomial algorithms, the first is based on the construction of a matroïd on this graph, and the second is to calculate in polynomial time a maximum matching with minimum cost on an induced graph constructed on the initial one. We prove that this two methods have the same result which is disjoints connexe unicyclic components with minimum total cost. In a second time, we are interesting by obtain a partition of unicyclic components with a minimum total cost by adding a girth constraint. Then the problem becomes difficult, and to solve it, we propose a polyhedral study in which we show many facets and valid inequalities. Some of them can be exactly separated in polynomial time. Then we solve the problem by a cutting plane algorithm based on these inequalities and using a compact formulation issued from the transversality of the bicircular matroïd. We also show a new valid inequality related to the upper bound of the number of unicyclic components obtained after the partition. The problem has practical and important applications in the design of telecommunication networks