Given an outerplane drawing  \(\mathcal {D}\) with n vertices, a \(\mathcal {D}\) -constrained maximum outerplane drawing is an outerplane drawing that contains \(\mathcal {D}\) and has the maximum number of edges among all such drawings. We show that (1) any such \(\mathcal {D}\) -constrained maximum outerplane drawing has at least \(n+1\) edges, and (2) this bound is best possible. If \(\mathcal {D}\) is a plane spanning path, we present an \(O(n^3)\) -time algorithm to compute a \(\mathcal {D}\) -constrained maximum outerplane drawing that minimizes its weight, that is, the sum of the Euclidean length of the edges of its drawing. For the unweighted setting, our results imply the following: It can be decided in \(O(n^3)\) time whether a given plane spanning path admits a polygon such that every edge of the path is on the polygon or an internal edge. Further, we present for several minimum-weight structures, like minimum spanning trees, points sets such that they are not subdrawings of the minimum-weight outerplane Laman graph of that point set.