The Kuo condition and the second Kuo condition are known as criteria for an r-jet to be V-sufficient in C-r mappings and Cr+1 mappings, respectively. In [2] we considered the notions of V-sufficiency of jets and these Kuo conditions in the relative case to a given closed set Sigma, and showed that the relative Kuo condition is a criterion for a relative r-jet to be Sigma-V-sufficient in C-r mappings and the second relative Kuo condition is a sufficient condition for a relative r-jet to be Sigma-V-sufficient in Cr+1 mappings. In this paper we discuss several conditions equivalent to the relative Kuo condition or the second relative Kuo condition. (c) 2021 Elsevier B.V. All rights reserved.