Strongly connected spanning subgraphs with the minimum number of arcs in semicomplete multipartite digraphs

We consider the problem (MSSS) of finding the minimum number of arcs in a spanning strongly connected subgraph of a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the hamiltonian cycle problem. We characterize the number of arcs in a minimum spanning strong subgraph for digraphs which are either extended semicomplete or semicomplete bipartite. Our proofs lead to polynomial algorithms for finding an optimal subgraph for every digraph from each of these classes. Our proofs are based on a number of results (some of which are new and interesting in their own right) on the structure of cycles and paths in these graphs. We conjecture that the problem is also polynomial for semicomplete multipartite digraphs, a superclass of the two classes considered. Quite recently, it was shown that the hamiltonian cycle problem is polynomially solvable for this class [?].