We study the $k$-center clustering problem under demographic fairness constraints, where the point set is partitioned into groups, and the aim is to compute clusters that exhibit a given group proportion. Previous work in this direction assumes that the entire point set already respects the desired proportions or uses relaxed notions of fairness. In this work, we propose a model that facilitates the creation of clusters that exactly match given target ratios, even when the input point set does not. We combine the well-known fair clustering model initiated by Chierichetti, Kumar, Lattanzi, and Vassilvitskii (NeurIPS 2017) with the notion of outliers to obtain a practical combinatorial framework that provides constant-factor approximate solutions for all proportion settings from $1:1$ for two groups to $t_1:t_2:\ldots:t_m$ for $m\geq 2$ groups, where $t_1,\ldots,t_m$ are integers. We implement and evaluate our algorithms, compare different variants, and provide evidence of the practicability of this approach.
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