Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n$n$-vertex graphs having a tree-decomposition into bags of diameter at most δ$\delta$, i.e., the tree-length δ$\delta$ graphs. For such graphs we construct additive 2δ$2\delta$-spanners with O(δn+nlogn)$O(\delta n+nlogn)$ edges, and additive 4δ$4\delta$-spanners with O(δn)$O\left(\delta n\right)$ edges. This provides new upper bounds for chordal graphs for which δ=1$\delta =1$. We also show a lower bound, and prove that there are graphs of tree-length δ$\delta$ for which every multiplicative δ$\delta$-spanner (and thus every additive (δ−1)$(\delta -1)$-spanner) requires Ω(n^1+1/Θ(δ))$\Omega \left(n^{1+1/\Theta \left(\delta \right)}\right)$ edges.