  In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We
also prove two relative index theorems for partitioned manifolds by using its
cyclic cocycle, which are generalizations of index theorems on partitioned
manifolds. One of these theorems is a variant of [M. Karami-A.H.S. Sadegh-M.E.
Zadeh, arXiv:1411.6090, Theorem 3.3].
