Interpolation conditions on recursive subdivision surfaces provide more powerful techniques to manipulate such surfaces. Recently, such conditions were extended to handle interpolation of pre-defined curves by a subdivision surface. Given a curve $C_i$ defined by a control polygon $cp_0$, this consists of constructing a strip complex $P_i$ as part of the defining polyhedral network $M_0$ or its first subdivision $M_1$. By repeated subdivision, $M_i$ converges to a limit surface $S$ which interpolates the curve $C_i$. In this paper, we describe an algorithm for constructing strip complexes that interpolate intersecting curves at the boundary of a surface. The algorithm is an important step towards solving the problem of interpolating arbitrary intersecting meshes of curves by subdivision surfaces.