Characterization of minimum cycle basis in weighted partial 2-trees

For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis (Liu and Lu, 2010). In our work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth at most two), which is a superclass of outerplanar graphs. In general graphs, a minimum cycle basis is a subset of the set of lex short cycles, where as the equality is known to hold only for the weighted outerplanar graphs.