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The device-to-device communication-aided fog radio access network, referred to as \textit{D2D-aided F-RAN}, takes advantage of caching at enhanced remote radio heads (eRRHs) and D2D proximity for improved system performance. For D2D-aided F-RAN, we develop a framework that exploits the cached contents at eRRHs, their transmission rates/powers, and previously received contents by different users to deliver the requesting contents to users with a minimum completion time. Given the intractability of the completion time minimization problem, we formulate it at each transmission by approximating the completion time and decoupling it into two subproblems. In the first subproblem, we minimize the possible completion time in eRRH downlink transmissions, while in the second subproblem, we maximize the number of users to be scheduled on D2D links. We design two theoretical graphs, namely \textit{interference-aware instantly decodable network coding (IA-IDNC)} and \textit{D2D conflict} graphs to reformulate two subproblems as maximum weight clique and maximum independent set problems, respectively. Using these graphs, we heuristically develop joint and coordinated scheduling approaches. Through extensive simulation results, we demonstrate the effectiveness of the proposed schemes against existing baseline schemes. Simulation results show that the proposed two approaches achieve a considerable performance gain in terms of the completion time minimization.