The $p$-center problem is to locate $p$ facilities in a network so as to minimize the longest distance between a demand point and its nearest facility. In this paper, we give a construction on a graph $G$ which produces an infinite ascending chain $G=G_0 \leq G_1 \leq G_2 \leq ...$ of graphs containing $G$ such that given any optimal solution $X$ for the $p$-center problem on $G$, $X$ is an optimal solution for the $p$-center problem on $G_i$ for any $i \geq 1$.