On Graphs Attaining Upper Bound of k-Rainbow Total Domination Number and Its Application in Graph Dynamical Systems

For a simple graph G with vertex set V(G),  and edge set a function \(f:V(G)\longrightarrow 2^{[k]}\) is a \(k-\) rainbow dominating function of G if for each vertex v in G for which \(f(v)=\emptyset ,\) \(\cup _{u\in N(v)}f(u)=[k].\) The weight of a \(k-\) rainbow dominating function is given by \(\Vert f\Vert =\sum _{v\in V(G)}|f(v)|.\) The \(k-\) rainbow domination number \(\gamma _{rk}(G)\) of G is the minimum over the weights of \(k-\) rainbow dominating functions of G. A \(k-\) rainbow dominating function f is said to be a \(k-\) rainbow total dominating function if the image of a vertex v under f is a singleton set then its entry should belong to the image of at least one vertex from the neighbourhood of v. The \(k-\) rainbow total domination number denoted by \(\gamma _{krt}(G)\) of G is the minimum over the weights of \(k-\) rainbow total dominating functions of G. In this article we explore the concept of \(k-\) rainbow total domination number in caterpillar trees and various other class of graphs. We also identified some class of graphs for which \(\gamma _{krt}(G)=k\gamma (G)\) , uncovering connection between these graph parameters and application of k-rainbow total domination number related to graph dynamical systems and domination in graph is discussed.