Summary: A fundamental notion in the theory of 2-structures is that of a primitive 2-structure. A. Ehrenfeucht and G. Rozenberg [Theor. Comput. Sci. 70, No. 3, 343-358 (1990; Zbl 0701.05053)]proved that primitivity is hereditary in the sense that each primitive 2-structure on \(n\) elements, where \(n \geq 3\), contains a primitive substructure on either \(n - 1\) or \(n - 2\) elements. In this paper we determine the class of primitive 2-structures on \(n\) elements that do not contain primitive substructures on \(n - 1\) elements: these 2-structures are said to satisfy the \((n - 2)\)-property. We show that for each \(n > 3\), there is a restricted number of primitive 2-structures on \(n\) elements satisfying this property In fact, for each \(n > 4\), there are four different reversible 2-structures up to isomorphism, satisfying the \((n - 2)\)- property, while for \(n\) odd, there are five different 2-structures with this property.