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 seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time O(m∆), where ∆ is the maximum degree of the graph with m edges. We use O(m∆ log∆) 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]). Department of Computer Science, University of Liverpool, Ashton Bldg, Liverpool L69 3BX, UK. Supported by EPSRC grant EP/F020651/1. Lab for Foundations of Computer Science, School of Informatics, University of Edinburgh, Edinburgh EH8 9AB Scotland, UK. Supported by EPSRC grant EP/D043905/1.