Abstract
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.
Highlights
A cycle basis of a graph G is a minimum-cardinality set C of cycles in G such that every cycle C in G can be written as the exclusive-or sum of a subset of cycles in C
A minimum cycle basis (MCB) of G is a cycle basis that minimizes the total weight of the cycles in the basis
By Lemma 6 it would suffice to compute the set of lex short cycles in G for our purposes. This is the approach of Liu and Lu, who showed that for outerplanar graphs an implicit representation of LSC(G) can be computed in linear time
Summary
A cycle basis of a graph G is a minimum-cardinality set C of cycles in G such that every cycle C in G can be written as the exclusive-or sum of a subset of cycles in C. We refer the interested reader to [16] for an exhaustive survey. It is—both from a practical and a theoretical viewpoint—an interesting task to compute minimum cycle bases efficiently. All graphs considered in this work are simple graphs G = (V, E) with a non-negative edge-weight function w : E → R≥0. (Computing MCBs for graphs with cycles of negative weight is an NP-hard problem [16]. In all previous work that we are aware of it is assumed that the edge-weights are non-negative.) Throughout this work, m = |E| denotes the size of the edge set and n = |V | the size of the vertex set of G
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