For any constants d\ge 1, \epsilon >0, t>1, and any n-point set P\subsetR^d, we show that there is a geometric graph G=(P,E) having O(n\log ^4 n\log\log n) edges with the following property: For any F\subseteq P, there exists F^+\supseteq F, |F^+| \le (1+\epsilon)|F| such that, for any pair p,q\in P\setminus F^+, the graph G-F contains a path from p to q whose (Euclidean) length is at most t times the Euclidean distance between p and q. In the terminology of robust spanners (Bose %DIFDELCMD < \etal%%% \ \et  2013) the graph G is a (1+\epsilon)k-robust t-spanner of P. This construction is more sparse than the most recent work ( Buchin, Ol\`ah, and Har-Peled 2018) which proves the existence of (1+\epsilon)k-robust t-spanners with n\log^{O(d)} n edges.
For any constants d\ge 1, \epsilon >0, t>1, and any n-point set P\subsetR^d, we show that there is a geometric graph G=(P,E) having O(n\log ^2 n\log\log n) edges with the following property: For any F\subseteq P, there exists F^+\supseteq F, |F^+| \le (1+\epsilon)|F| such that, for any pair p,q\in P\setminus F^+, the graph G-F contains a path from p to q whose (Euclidean) length is at most t times the Euclidean distance between p and q. In the terminology of robust spanners (Bose %DIFDELCMD < \etal%%% \et al, SICOMP, 42(4):1720--1736, 2013) the graph G is a (1+\epsilon)k-robust t-spanner of P. This construction is sparser than the recent constructions of Buchin, Ol\`ah, and Har-Peled (arXiv:1811.06898) who prove the existence of (1+\epsilon)k-robust t-spanners with n\log^{O(d)} n edges.
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