  We give a simple polynomial-time algorithm to exactly count the number of
Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show
how to adapt this algorithm to exactly sample a random ET of the given
generalized series-parallel graph. Note that the class of generalized
seriesparallel graphs includes all outerplanar graphs. We can perform the
counting in time $O(m\Delta^3)$, where $\Delta$ is the maximum degree of the
graph with $m$ edges. We use $O(m\Delta^2 \log \Delta)$ bits to store
intermediate values during our computations. To date, these are the first known
polynomial-time algorithms to count or sample ETs of any class of graphs; there
are no other known polynomial-time algorithms to even approximately count or
sample ETs of any other class of graphs. The problem of counting ETs is known
to be $#P$-complete for general graphs (Brightwell and Winkler, 2005 [3]) and
also for planar graphs (Creed, 2009 [4]).
