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Given a strongly connected directed graph D, let SD denote the set of all stochastic matrices whose directed graph is a spanning subgraph of D. We consider the problem of completely describing the set of stationary vectors of irreducible members of SD. Results from the area of convex polytopes and an association of each matrix with an undirected bipartite graph are used to derive conditions which must be satisfied by a positive probability vector x in order for it to be admissible as a stationary vector of some matrix in SD. Given some admissible vector x, the set of matrices in SD that possess x as a stationary vector is also characterised.
Breen et al. (Tue,) studied this question.