Tree decompositions of small diameter

Abstract

Motivated by applications in parallel and dynamic graph algorithms, we investigate the tradeoff between width and diameter of tree decompositions. For all integers n, k and K with 1 ≤ k ≤ K ≤ n− 1, denote by D(n, k, K) the maximum, over all n-vertex graphs G of treewidth k, of the smallest diameter of a tree decomposition of G of width K. We determine D(n, k, K), up to a constant factor, for all values of n, k and K. When K is bounded by a constant (the case of greatest practical relevance), D(n, k, K) is θ(n) for K ≤ 2k-1, θ(√n) for 2k ≤ K ≤ 3k−2, and θ(log n) for K ≥ 3k−1. We provide much more accurate bounds for the case K ≤ 2k−1.

This research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM-IT: Algorithms and Complexity in Information Technology). The work was carried out while the first author was with the Max-Planck-Institut für Informatik in Saarbrücken, Germany.