This paper addresses the problem of efficiently selecting sensors such that the mean squared estimation error is minimized under jointly Gaussian assumptions. First, we propose an O(n3) algorithm that yields the same set of sensors as a previously published near mean squared error (MSE) optimal method that runs in O(n4). Then we show that this approach can be extended to efficient sensor selection in a time varying graph. We consider a rank one modification to the graph Laplacian, which captures the cases where a new edge is added or deleted, or an edge weight is changed, for a fixed set of vertices. We show that we can efficiently update the new set of sensors in O(n2) time for the best case by saving computations that were done for the original graph. Experiments demonstrate advantages in computational time and MSE accuracy in the proposed methods compared to recently developed graph sampling methods.