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 (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].