Let n be any positive integer and F be a family of subsets of [n]. A family F′ is said to be D-secting for F if for every A∈F, there exists a subset A′∈F′ such that |A∩A′|−|A∩([n]∖A′)|=i, where i∈D, D⊆{−n,−n+1,…,0,…,n}. A D-secting family F′ of F, where D={−1,0,1}, is a bisecting family ensuring the existence of a subset A′∈F′ such that |A∩A′|∈{⌈[Formula presented]⌉,⌊[Formula presented]⌋}, for each A∈F. In this paper, we study D-secting families for F with restrictions on D, and the cardinalities of F and the subsets of F. © 2017 Elsevier B.V.