Abstract

This paper deals with the length of a Robertson–Seymour's tree-decomposition. The tree-length of a graph is the largest distance between two vertices of a bag of a tree-decomposition, minimized over all tree-decompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded tree-length graphs includes (but is not reduced to) bounded chordality graphs (like interval graphs, permutation graphs, AT-free graphs, etc.). For instance, we show that the tree-length of any outerplanar graph is k/3, where k is the chordality of the graph, and we compute the tree-length of meshes.

More fundamentally we show that any algorithm computing a tree-decomposition approximating the tree-width (or the tree-length) of an n-vertex graph by a factor α or less does not give an α-approximation of the tree-length (resp. the tree-width) unless if α=Ω(n^1/5). We complete these results presenting several polynomial time constant approximate algorithms for the tree-length.

The introduction of this parameter is motivated by the design of compact distance labeling, compact routing tables with near-optimal route length, and by the construction of sparse additive spanners.

Keywords: Tree-decomposition; Tree-width; Tree-length; Chordality; Small separator

Article Outline

1. Introduction

1.1. Our results

1.2. Definition

2. Optimal length tree-decomposition

2.1. Preliminary results

2.2. Outerplanar graphs

2.3. Tree-length of meshes

2.4. Width-length trade-off

2.5. Specific tree-decompositions

3. Approximation algorithm

3.1. Algorithm LexM

3.2. Algorithm BFS-Layering

3.3. An heuristic: disk-tree

4. Conclusion and further works

References