Facets of the (s,t)-p(s,t)-p-path polytope

We give a partial description of the (s,t)-p(s,t)-p-path polytope of a directed graph DD which is the convex hull of the incidence vectors of simple directed (s,t)(s,t)-paths in DD of length pp. First, we point out how the (s,t)-p(s,t)-p-path polytope is located in the family of path and cycle polyhedra. Next, we give some classes of valid inequalities which are very similar to the inequalities which are valid for the pp-cycle polytope, that is, the convex hull of the incidence vectors of simple cycles of length pp in DD. We give necessary and sufficient conditions for these inequalities to be facet defining. Furthermore, we consider a class of inequalities that has been identified to be valid for (s,t)(s,t)-paths of cardinality at most pp. Finally, we transfer the results to related polytopes, in particular, the undirected counterpart of the (s,t)-p(s,t)-p-path polytope.