Before revision:  We also prove that E(C') \leq E(C)  +3 if G does not contain faces of size five, which gives a new tool for proving results about cycle spectra and evidence that these face sizes obstruct long cycles .
After revision:  We also prove that E(C') \leq E(C) +3 if G contains no face of size five, which gives a new tool for proving results about cycle spectra and evidence that these face sizes obstruct long cycles .