The family Pk of graphs with proper-path-width at most k is minor-closed. It is known that the number of minimal forbidden minors for a minor-closed family of graphs is finite, but we have few such families for which all the minimal forbidden minors are listed. Although the minimal acyclic forbidden minors are characterized for Pk, all the minimal forbidden minors are known only for P1. This paper lists 36 minimal forbidden minors for P2, and shows that there exist no other minimal forbidden minors for P2.

We introduce the interval set of a graph G which is a representation of the proper-path-decomposition of G, and show a linear time algorithm to construct an optimal interval set for any tree T. It is shown that a proper-path-decomposition of T with optimal width can be obtained from an optimal interval set of T in O(n log n) time.