A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, given a biconnected planar graph G with fixed plane embedding and a subset A of corners of G, we consider the problem of finding a star-shaped drawing D of G such that only corners in A are allowed to become concave corners in D. We first characterize a necessary and sufficient condition for a subset A of corners to admit such a star-shaped drawing D. Then we present a linear time algorithm for finding such a star-shaped drawing D. Our characterization includes Thomassen's classical characterization of biconnected plane graphs with a prescribed boundary that have convex drawings.