This paper presents a novel time-optimal motion planning strategy for a mobile robot with kinematic constraints. The method works in environments in presence of obstacles, without needing to generate the configuration space for the robot. Further, it derives a minimum time first derivative smooth path, as opposed to a minimum distance path which is commonly given by various present solution techniques. The problem is solved in three stages: (i) A reduced visibility graph for a point object is obtained. (ii) The reduced visibility graph is converted into a feasible reduced visibility graph accounting for the size and kinematic constraints of the robot. (iii) The A* algorithm is used to search the feasible reduced visibility graph with the cost function being the time of travel, to obtain a safe, time-optimal, smooth path. The algorithm runs in polynomial time. The method has been tested in computer simulations and test results are presented