In this paper, we propose a control method for leader-follower navigation in obstacle environments while preserving sensing network connectivity without data transmission between robots. Unlike most connectivity-preserving algorithms, the control input is determined in such a way as to not only guarantee connectivity preservation and collision avoidance, but also to ensure that input constraints are not violated at each time step. We also introduce a simple rule for changing network topology depending on environments such that some sensing links are deactivated in order to pass through narrow spaces, while active links are increased in free spaces to keep the group as cohesive as possible. The effectiveness of the proposed method is demonstrated in simulations and experiments.