Elsevier

Discrete Mathematics

Volume 307, Issue 16, 28 July 2007, Pages 2008-2029
Discrete Mathematics

Tree-decompositions with bags of small diameter

https://doi.org/10.1016/j.disc.2005.12.060Get rights and content
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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(n1/5). 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

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Work partially supported by the Research and Training Network COMBSTRU (HPRN-CT-2002-00278).