$\mathcal{O}(k)$-robust spanners in one dimension

A geometric $t$-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most $t$ times the Euclidean distance between the points. Informally, a spanner is $\mathcal{O}(k)$-robust if deleting $k$ vertices only harms $\mathcal{O}(k)$ other vertices. We show that on any one-dimensional set of $n$ points, for any $\varepsilon>0$, there exists an $\mathcal{O}(k)$-robust $1$-spanner with $\mathcal{O}(n^{1+\varepsilon})$ edges. Previously it was only known that $\mathcal{O}(k)$-robust spanners with $\mathcal{O}(n^2)$ edges exists and that there are point sets on which any $\mathcal{O}(k)$-robust spanner has $\Omega(n\log{n})$ edges.
Comment: 6 pages, 6 figures