Computing minimum cycle bases in weighted partial 2-trees in linear time

We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis in weighted partial 2-trees (i.e., graphs of treewidth at most two) with non-negative edge-weights. The implicit representation can be made explicit in a running time that is proportional to the size of the minimum cycle basis. For planar graphs, Borradaile, Sankowski, and Wulff-Nilsen [Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time, FOCS 2010] showed how to compute an implicit O(n log n) space representation of an minimum cycle basis in O(n log5 n) time. For the special case of par-tial 2-trees, our algorithm improves this result to linear time and space. 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].