Journal of Graph Algorithms and Applications
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Computing Minimum Cycle Bases in Weighted Partial 2-Trees in Linear Time
Vol. 18, no. 3, pp. 325-346, 2014. Regular paper.
Abstract 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 logn) space representation of an minimum cycle basis in O(n log5 n) time. For the special case of partial 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]
Submitted: July 2013.
Reviewed: October 2013.
Revised: January 2014.
Reviewed: April 2014.
Revised: April 2014.
Accepted: May 2014.
Final: May 2014.
Published: June 2014.
Communicated by Stephen G. Kobourov