A Novel Conditional Connectivity Measure: k-Total Domination Edge Connectivity of Graphs

In recent years, domination-based connectivity concepts have attracted growing interest as extensions of conditional connectivity in graph theory. In this study, we introduce a new conditional connectivity parameter that incorporates the notion of total domination, namely the k-total domination edge connectivity. Formally, for a connected graph G=(V,E), the parameter is defined as the smallest number of edges whose removal disconnects the graph in such a way that each resulting component has a total domination number equal to k. We particularly investigate the case k=2 and provide explicit calculations of this measure for fundamental graph classes such as paths, cycles, and complete graphs. The proposed approach aims to contribute to a deeper understanding of network robustness under domination-based constraints.