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Multi-kernel learning (MKL) has been widely used in function approximation tasks. The key problem of MKL is to combine kernels in a prescribed dictionary. Inclusion of irrelevant kernels in the dictionary can deteriorate accuracy of MKL, and increase the computational complexity. To improve the accuracy of function approximation and reduce the computational complexity, the present paper studies data-driven selection of kernels from the dictionary that provide satisfactory function approximations. Specifically, based on the similarities among kernels, the novel framework constructs and refines a graph to assist choosing a subset of kernels. In addition, random feature approximation is utilized to enable online implementation for sequentially obtained data. Theoretical analysis shows that our proposed algorithms enjoy tighter sub-linear regret bound compared with state-of-art graph-based online MKL alternatives. Experiments on a number of real datasets also showcase the advantages of our novel graph-aided framework., Comment: Preliminary results of this work have been presented in "Online Multi-Kernel Learning with Graph-Structured Feedback." P. M. Ghari, and Y. Shen, International Conference on Machine Learning (ICML), pp. 3474-3483. PMLR, July 2020