Summary: Let \(G=(V,E)\) be an undirected connected graph with positive edge lengths. The vertex \(p\)-center problem is to find the optimal location of \(p\) centers so that the maximum distance to a vertex from its nearest center is minimized, where the centers may be placed at the vertices. \textit{O. Kariv} and \textit{S. L. Hakimi} [SIAM J. Appl. Math. 37, 513-538 (1979; Zbl 0432.90074)] have shown that this problem is NP-hard. We will consider two modifications of this problem in which each center may be located in one of two predetermined vertices. We will show the NP- completeness of their recognition versions.