Genus Distributions of cubic series-parallel graphs
  We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3. 
Jonathan L. Gross, Michal Kotrb\v{c}\'ik, Timothy Sun
cs.DM math.CO