Characterization of Minimum Cycle Basis in Weighted Partial 2-trees

TL;DR

This paper proves that in weighted partial 2-trees, the set of lex short cycles forms a minimum cycle basis, extending known results from outerplanar graphs to a broader class.

Contribution

It establishes that lex short cycles constitute a minimum cycle basis in weighted partial 2-trees, broadening the understanding of cycle bases in graph theory.

Findings

Lex short cycles form a minimum cycle basis in weighted partial 2-trees.

Extension of previous results from outerplanar graphs to partial 2-trees.

Provides theoretical proof for the characterization of cycle bases in this class.

Abstract

For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis [Inf. Process. Lett. 110 (2010) 970-974 ]. In this work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth two) which is a superclass of outerplanar graphs.