Let \(\mathbb C\) be a pointed finitely complete category and let \(S\) be a class of objects in \(\mathbb C\). Denote by \(\mathrm{Pt }\mathbb C\) the category of points in \(\mathbb C\), i.e., pairs of morphisms \((f,s)\) such that \(fs=1\), and by \(\mathrm{SPt }\mathbb C\) the full subcategory of \(\mathrm{Pt }\mathbb C\) whose objects are those which are in \(S\). In this paper, the authors introduce the notion of \(S\)-protomodular category with respect to a suitable class \(S\) of points, stable under pullbacks, in the following way: \(\mathbb C\) is \(S\)-protomodular if any object in \(\mathrm{SPt }\mathbb C\) is a strong point, in the sense of D. Bourn [Theory Appl. Categ. 28, 150–65 (2013; Zbl 1273.18007)] (see also the paper of N. Martins-Ferreira et al. [Appl. Categ. Struct. 22, No. 5–6, 687–697 (2014; Zbl 1311.18005)]), and \(\mathrm{SPt }\mathbb C\) is closed under finite limits in \(\mathrm{Pt }\mathbb C\). For example, monoids with operations and the class of Schreier points are examples of this new notion of protomodular category. On the other hand, \(S\)-reflexive graphs and \(S\)-reflexive relations are the bridge between \(S\)-protomodular and Mal’tsev categories. In Section 6 the authors introduce the notion of \(S\)-special morphism and, in Section 7, the notion of protomodular core of an \(S\)-protomodular category. Using these definitions, it is possible to characterize internal groupoids among internal \(S\)-categories and equivalence relations among \(S\)-reflexive relations. Morever, many partial aspects of Mal’tsev and protomodular categories can be obtained in the \(S\)-protomodular setting. For example, the ones related with the centrality of reflexive relations.