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In this paper, we study reducible surgeries on knots in $S^3$. We developthickness bounds for L-space knots that admit reducible surgeries, and lowerbounds on the slice genus for general knots that admit reducible surgeries. TheL-space knot thickness bounds allow us to finish off the verification of theCabling Conjecture for thin knots, which was mostly worked out in\cite{DeY21b}. We also provide a new upper bound on reducing slopes forfibered, hyperbolic slice knots and on multiple reducing slopes for sliceknots. Our techniques involve the $d$-invariants and mapping cone formula fromHeegaard Floer homology.

Obstructing Reducible Surgeries: Slice Genus and Thickness Bounds