Let G be an embedded planar undirected graph that has n vertices, m edges,
and f faces but has no self-loop or multiple edge. If G is triangulated, we
can encode it using 4/3 m 1 bits, improving on the best previous bound of
about 1.53m bits. In case exponential time is acceptable, roughly 1.08m bit
s have been known to suffice. If G is triconnected, we use at most (2.5 + 2
log 3) min {n,f} 7 bits, which is at most 2.835m bits and smaller than the
best previous bound of 3m bits. Both of our schemes take O(n) time for enc
oding and decoding.