  There are many representations of planar graphs, but few are as elegant as
Tur\'an's (1984): it is simple and practical, uses only 4 bits per edge, can
handle self-loops and multi-edges, and can store any specified embedding. Its
main disadvantage has been that "it does not allow efficient searching"
(Jacobson, 1989). In this paper we show how to add a sublinear number of bits
to Tur\'an's representation such that it supports fast navigation while
retaining simplicity. As a consequence of the inherited simplicity, we offer
the first efficient parallel construction of a compact encoding of a planar
graph embedding. Our experimental results show that the resulting
representation uses about 6 bits per edge in practice, supports basic
navigation operations within a few microseconds, and can be built sequentially
at a rate below 1 microsecond per edge, featuring a linear speedup with a
parallel efficiency around 50\% for large datasets.
