  We present a linear time algorithm for computing an implicit linear space
representation of a minimum cycle basis (MCB) in weighted partial 2-trees,
i.e., graphs of treewidth two. The implicit representation can be made explicit
in a running time that is proportional to the size of the MCB.
  Our algorithm improves the result of Borradaile, Sankowski, and Wulff-Nilsen
[Min $st$-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time,
FOCS 2010]---which computes for all planar graphs an implicit $O(n \log n)$
space representation of an MCB in $O(n \log^5 n)$ time---by a polylog factor
for the special case of partial 2-trees. Such an improvement was achieved
previously only for outerplanar graphs [Liu and Lu: Minimum Cycle Bases of
Weighted Outerplanar Graphs, IPL 110:970--974, 2010].
