Aldred and Plummer (1999) have proved that every m-connected K-1,K-m-k+2-free graph of even order contains a perfect matching which avoids k prescribed edges. They have also proved that the result is best possible in the range 1 <= k <= 1/2(m + 1). In this paper, we show that if 1/2(m + 2) <= k <= m - 1, their result is not best possible. We prove that if m >= 4 and 1/2(m + 2) <= k <= m - 1, every K-1,K-[2m-k+4/3]-free graph of even order contains a perfect matching which avoids k prescribed edges. While this is a best possible result in terms of the order of a forbidden star, if 2m - k + 4 equivalent to 0 (mod 3), we also prove that only finitely many sharpness examples exist. (C) 2015 Elsevier B.V. All rights reserved.