Exact counting of Euler tours for generalized series-parallel graphs

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 series-parallel graphs includes all outerplanar graphs. We can perform the counting in time O(mΔ³), where Δ is the maximum degree of the graph with m edges. We use O(mnΔ²logΔ) 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 [2]) also for planar graphs (Creed, 2010 [3]).