Status split and nonsplit domination in graphs

Network connectivity, which measures whether the network stays connected or disconnected after the removal of the most critical set of nodes, can be employed to determine the robustness and stability of a network. Two adjacent nodes u and v, in a connected graph [Formula: see text] “mutually status unequal (stu) dominate” one another if their status values, as determined by a distance parameter (such as the shortest path), are unequal. By way of implementation, a novel idea known as the stu-split stu-dominanting set, consists of critical nodes, the removal of which splits the graph network [Formula: see text] has been devised. In parallel, stu-nonsplit dominanting set is defined so that [Formula: see text] is still connected even after the critical nodes of this set are removed. Furthermore, stu-split and stu-nonsplit stu-domination numbers for some classes of graphs are obtained. Also, minimality condition of both the sets has been investigated.