Pathlength of Outerplanar Graphs

Abstract

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 $p\ell (G)$, 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 $c<\frac{3}{2}$ unless $P=NP$). In this work, we focus on the study of the pathlength of simple sub-classes of planar graphs. We start by designing a linear-time algorithm that computes the pathlength of trees. Then, we show that the pathlength of cycles with n vertices is equal to $\lfloor \frac{n}{2}\rfloor$. 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 $p\ell (G)+1$) path-decompositions of outerplanar graphs.

This work is partially founded by projects UCA JEDI (ANR-15-IDEX-01), the ANR project Digraphs (ANR-19-CE48-001) and EUR DS4H (ANR-17-EURE-004) Investments in the Future. Due to lack of space, some proofs are omitted or sketched. Full proofs can be found in [5].