Abstract
For each non-negative integer k, we provide all outerplanar obstructions for the class of graphs whose cycle matroid has pathwidth at most k. Our proof combines a decomposition lemma for proving lower bounds on matroid pathwidth and a relation between matroid pathwidth and linear width. Our results imply the existence of a linear algorithm that, given an outerplanar graph, outputs its matroid pathwidth.
Highlights
The notions of pathwidth and branchwidth are fundamental graph parameters that appear in many topics of discrete mathematics and algorithms
We study the set obs(Pk) and we characterize, for every k, all members of obs(Pk) that are cycle matroids of outerplanar graphs
We show that several structural characteristics of the pathwidth of acyclic graphs are transferred to the μ-pathwidth of outerplanar graphs
Summary
The notions of pathwidth and branchwidth are fundamental graph parameters that appear in many topics of discrete mathematics and algorithms. The pathwidth of a matroid was defined by Geelen, Gerards, and Whittle in [7] (see [9]) and was extensively studied in the work of Kashyap [13] in the context of trellis state-complexity of linear codes. We prove the existence of a bijection between acyclic obstructions for linear-width (a parameter very similar to the pathwidth for graphs) and the outerplanar obstructions of μ-pathwidth. This gives a precise characterization of all members of obs(Pk) that are cycle matroids of outerplanar graphs. A byproduct of our results is that the μ-pathwidth of outerplanar graphs can be computed in linear time
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