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(m3 ), where is the maximum degree of the graph with m edges. We use O(m2 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...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Prasad Chebolu, Mary Cryan, Russell A. Martin
• Keywords: CORR 2010, Education, Polynomial-time Algorithms, Series-parallel Graph, Simple Polynomial-time Algorithm