Treelength of Series-parallel Graphs

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