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In the Steiner point removal (SPR) problem, we are given a (weighted) graph$G$ and a subset $T$ of its vertices called terminals, and the goal is tocompute a (weighted) graph $H$ on $T$ that is a minor of $G$, such that thedistance between every pair of terminals is preserved to within some smallmultiplicative factor, that is called the stretch of $H$.  It has been shown that on general graphs we can achieve stretch $O(\log |T|)$[Filtser, 2018]. On the other hand, the best-known stretch lower bound is $8$[Chan-Xia-Konjevod-Richa, 2006], which holds even for trees. In this work, weshow an improved lower bound of $\tilde\Omega\big(\sqrt{\log |T|}\big)$.

An $\tilde\Omega\big(\sqrt{\log |T|}\big)$ Lower Bound for Steiner Point Removal