Paths and cycles in extended and decomposable digraphs

We consider digraphs — called extended locally semicomplete digraphs, or extended LSD's, for short — that can be obtained from locally semicomplete digraphs by substituting independent sets for vertices. We characterize Hamiltonian extended LSD's as well as extended LSD's containing Hamiltonian paths. These results as well as some additional ones imply polynomial algorithms for finding a longest path and a longest cycle in an extended LSD. Our characterization of Hamiltonian extended LSD's provides a partial solution to a problem posed by Häggkvist (1993). Combining results from this paper with some general results derived for the so-called totally Φ0-decomposable digraphs in Bang-Jensen and Gutin (1996) we prove that the longest path problem is polynomially solvable for totally Φ0-decomposable digraphs — a fairly wide family of digraphs which is a common generalization of acyclic digraphs, semicomplete multipartite digraphs, extended LSD's and quasi-transitive digraphs. Similar results are obtained for the longest cycle problem and other problems on cycles in subfamilies of totally Φ0-decomposable digraphs. These polynomial algorithms are a natural and fairly deep generalization of algorithms obtained for quasi-transitive digraphs in Bang-Jensen and Gutin (1996) in order to solve a problem posed by N. Alon