This article uses a vertex-closing approach to investigate the p-center problem. The optimal set of vertices to close are found in imbedded subgraphs of the original graph. Properties of these subgraphs are presented and then used to characterize the optimal solution, to establish a priori upper and lower bounds, to establish refined lower bounds, and to verify the optimality of solutions. These subgraphs form the foundation of two polynomial algorithms of complexity O(ℝEℝlogℝEℝ) and O(ℝEℝ2). The algorithms are proven to converge to an optimum for special cases, and computational evidence is provided which suggests that they produce very good solutions more generally. Both algorithms perform very well on problems where p is large relative to the number of vertices n, specifically, when p/n≥0.30. One of the algorithms is especially efficient for solving a sequence of problems on the same graph.
