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Treelength of Series-parallel graphs


The length of a tree-decompositionof a graph is the maximum distance between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decomposition. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of agraph cannot be approximated up to a factor less than3/2 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the maximum size of its isometric cycles. This work initiates the study of treelength in planar graphs by considering its next natural subclass, namely the one of series-parallel graphs. We first fully describe the treelength of melon graphs (set of pairwise internally disjointpaths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up toa factor 3/2 in series-parallel graphs. Our main result is a polynomial-time algorithm for deciding whether a series-parallel graph has treelength at most 2. Our latter result relies on a characterization of series-parallel graphs with treelength 2 in terms of an infinite family of forbidden isometric subgraphs.
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hal-03175837 , version 1 (21-03-2021)



Thomas Dissaux, Guillaume Ducoffe, Nicolas Nisse, Simon Nivelle. Treelength of Series-parallel graphs. LAGOS 2021 - XI Latin and American Algorithms, Graphs and Optimization Symposium, May 2021, São Paulo / Virtual, Brazil. ⟨10.1016/j.procs.2021.11.008⟩. ⟨hal-03175837⟩
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