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We propose a joint optimization model for capacity design of networks with p-cycles. The model is based on a modified definition of network fundamental cycles and the available straddling links. Concepts about visible and hidden straddling links, which are essential components of our model are also introduced. This is the first ILP model for joint optimization of p-cycle network that can be solved without enumerating p-cycle candidates, and has the ability to achieve optimum solutions. In addition, the complexity of our proposed model is much smaller than any conventional models, particularly when applying to a planar network. This model is suitable large size networks and for shared risk link group networks or backbone networks protected by p-cycle schemes.