Pathlength of Outerplanar graphs

A path-decomposition of a graph G = (V, E) is a sequence of subsets of V , called bags, that satisfy some connectivity properties. The length of a path-decomposition of a graph G is the greatest distance between two vertices that belong to a same bag and the pathlength, denoted by pl(G), of G is the smallest length of its path-decompositions. This parameter has been studied for its algorithmic applications for several classical metric problems like the minimum eccentricity shortest path problem, the line-distortion problem, etc. However, deciding if the pathlength of a graph G is at most 2 is NP-complete, and the best known approximation algorithm has a ratio 2 (there is no c-approximation with c < 3/2). In this work, we focus on the study of the pathlength of simple sub-classes of planar graphs. We start with the pathlength of cycles with n vertices which is equal to n/2. Then, we design a lineartime algorithm that computes the pathlength of trees. Finally, our main result is a (+1)approximation algorithm for the pathlength of outerplanar graphs. This algorithm is based on a characterization of almost optimal (of length at most pl(G) + 1) path-decompositions of outerplanar graphs