Given a graph G =(V,E) and an integer value f(v) for each node v in V, a node subset D is a total f-dominating set if every node v is adjacent to at least f(v) nodes of D. Given a weight c(v) for each node v, the minimum weight total f-dominating set problem is to find a total f-dominating set of minimum total weight. In this article, we propose a polyhedral study of the associated polytope together with a complete and compact description of the polytope for totally unimodular graphs and cycles. We also propose a linear time dynamic programming algorithm for the case of trees