Remote clouds are gradually unable to achieve ultra-low latency to meet the requirements of mobile users because of the intolerable long distance between remote clouds and mobile users and the network congestion caused by the tremendous number of users. Mobile edge computing, a new paradigm, has been proposed to mitigate aforementioned effects. Existing studies mostly assume the edge servers have been deployed properly and they just pay attention to how to minimize the delay between edge servers and mobile users. In this paper, considering the practical environment, we investigate how to deploy edge servers effectively and economically in wireless metropolitan area networks. Thus, we address the problem of minimizing the number of edge servers while ensuring some QoS requirements. Aiming at more consistence with a generalized condition, we extend the definition of the dominating set, and transform the addressed problem into the minimum dominating set problem in graph theory. In addition, two conditions are considered for the capacities of edge servers: one is that the capacities of edge servers can be configured on demand, and the other is that all the edge servers have the same capacities. For the on-demand condition, a greedy based algorithm is proposed to find the solution, and the key idea is to iteratively choose nodes that can connect as many other nodes as possible under the delay, degree and cluster size constraints. Furthermore, a simulated annealing based approach is given for global optimization. For the second condition, a greedy based algorithm is also proposed to satisfy the capacity constraint of edge servers and minimize the number of edge servers simultaneously. The simulation results show that the proposed algorithms are feasible.

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