The localization of sources of pressure transients is essential for proactively managing and reducing the adverse effects of these transients in water supply networks (WSNs). This paper addresses the issue of localizing transient sources in a WSN where there is uncertainty about the network connectivity. If closed valves or blockages in the pipes are not accounted for, it can result in inaccurate knowledge of the network connectivity, leading to incorrect estimations about the sources of pressure transients. The problem is challenging due to the added uncertainties in estimating the velocity of pressure waves and determining their arrival times at pressure monitoring sites. In order to systematically investigate this problem, the paper presents a novel theoretical framework for localizing the source of pressure transients in WSNs with uncertain connectivity. The problem is formulated as a mixed-integer quadratic program, which consists of minimizing the difference between analytical and measured pressure wave arrival times for multiple pairs of time-synchronized sensors. Unlike previous approaches, the k-shortest path (with respect to time) routing problem is incorporated into the problem formulation to account for multiple potential wave propagation paths. The optimization problem is then solved using an off-the-shelf solver, and a methodology is developed to ensure the reliability of the optimization approach. We apply the proposed methodology to numerically simulated pressure transient data for a benchmark WSN and compare it against a previously published method. The results show a notable improvement in accurately localizing the source of a transient in the presence of unknown closed valves or pipe blockages. Provided pressure wave speeds are predetermined, the proposed methodology is able to simultaneously localize the source of a pressure transient and validate the assumed hydraulic connectivity of a WSN. In this way, any irregularities or uncertainties in network connectivity can also be periodically detected and validated.