Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most δ, i.e., the tree-length δ graphs. For such graphs we construct additive 2δ-spanners with O(δn+n log n) edges, and additive 4δ-spanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of tree-length δ for which every multiplicative δ-spanner (and thus every additive (δ − 1)-spanner) requires Ω(n1+1/Θ(δ)) edges. c © 2007 Elsevier B.V. All rights reserved.