Frames of cycles and directed cycles

Let n > m ≥ 3 be integers. The framing number fr(C_m, C_n) of a cycle C_m of length m and a cycle C_n of length n is defined as the minimum order of a graph every vertex of which belongs to an induced C_m and an induced C_n. The framing number fr ( C_m, C_n) of a directed cycle C_m of length m and a directed cycle C_n of length n is defined as the minimum order of a digraph every vertex of which belongs to an induced C_m and an induced C_n. In this paper we determine all the nonisomorphic frames of those pairs of cycles C_m and C_n which have framing number n + 2. Thereafter we characterize all those pairs of directed cycles C_m and C_n which have framing number n + 2 and, for each such pair (m, n), we determine all the nonisomorphic frames of C_m and C_n.