The P4-structure of a graph G is a hypergraph H on the same vertex set such that four vertices form a hyperedge in H whenever they induce a P4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.