  A permutation graph is a cubic graph admitting a 1-factor M whose complement
consists of two chordless cycles. Extending results of Ellingham and of
Goldwasser and Zhang, we prove that if e is an edge of M such that every
4-cycle containing an edge of M contains e, then e is contained in a
subdivision of the Petersen graph of a special type. In particular, if the
graph is cyclically 5-edge-connected, then every edge of M is contained in such
a subdivision. Our proof is based on a characterization of cographs in terms of
twin vertices. We infer a linear lower bound on the number of Petersen
subdivisions in a permutation graph with no 4-cycles, and give a construction
showing that this lower bound is tight up to a constant factor.
