The p-center problem on a network is a model to locate p new facilities that will serve n existing demand points on that network. The objective is to minimize the maximum of the weighted distances between each demand point and its nearest new facility. This type of problem usually arises in the location of emergency facilities like hospitals, police and fire stations. The problem is known to be VP-Hard on a cyclic network, but polynomial-time solvable on a tree network. In this study, a spanning tree approach to solving the problem on a cyclic network is discussed. First, the existence of an optimal spanning tree that gives the network optimal solution, is proved. Then, two specific types of spanning trees are introduced and experimentally tested whether they contain the optimal tree or not. Also, some properties of such an optimal tree are discussed and some special cases for which the optimal tree can be determined in polynomial time, are identified.