Aldred and Plummer (1999) have proved that every m-connected K1,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≤12(m+1). In this paper, we show that if 12(m+2)≤k≤m-1, their result is not best possible. We prove that if m≥4 and 12(m+2)≤k≤m-1, every K1,⌈2m-k+43⌉-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≡0(mod3), we also prove that only finitely many sharpness examples exist.
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