A uniquely colorable graph G whose chromatic partition contains atleast one g - set is termed as a g - uniquely colorable graph. In this paper, we provide necessary and sufficient condition for and G* to be g - uniquely colorable whenever G g- uniquely colorable and also provide constructive characterization to show that whenever G is g- uniquely colorable such that |P | ³ 2, G can be both planarand non planar.