A Note on the Polynomial D\(\begin{array}{ccc} v\\t \end{array}\)(G, x)

In various practical scenarios, especially in communication networks, sensor grids, and surveillance systems, it becomes essential to ensure that certain critical nodes (vertices) are included in every total dominating set of a graph. A total dominating set is a subset of vertices in a graph such that every vertex is adjacent to at least one vertex in this set. However, in specific applications, some nodes may serve as vital control hubs, data aggregators or monitoring stations that must be active or functional in any dominating configuration. In this paper the total domination polynomial D\(\begin{array}{ccc} v\\t \end{array}\)(G, x) in which a particular vertex v of G is present in every TD-set of G is determined for certain classes of graphs.