Given a graph G = (V,E) and a non negative cost function on edges, the Hop-constrained Path Problem (HPP) consists of finding between two distinguished vertices s and t of V a minimum cost path with no more than L edges where L is a fixed integer. Dahl characterised the dominant of the convex hull of the incidence vectors of st-paths of length bounded by L, denoted by DL(G), for any graph G when L ≤ 3, using trivial, st-cut, and L-path-cut inequalities. A graph G is said L-h-simple if the set of Dahl's inequalities is sufficient to define DL(G). In this paper, we study the L-h-simple property when L ≥ 4. We present some results on the facial structure of the dominant DL(G). We also examine some basic operations on graphs which preserve the L-h-simple property.