A Study on Total Rebellion Number in Graphs
V. Mohanaselvi, P. Shyamala Anto Mary, A. Monisha · Journal of Informatics and Mathematical Sciences · 2017
A set $R\subseteq V$ of a graph \(G = (V,E)\) is said to be a `rebellion set' (\(rb\)-set) of \(G\), if \(\arrowvert N_R(v) \arrowvert \leq \arrowvert N_{V/R}(v) \arrowvert\), for all \(v\in R\), \(\arrowvert R\arrowvert \geq \arrowvert V/R \arrowvert\) and \(\la R\ra\) has no isolated vertices. The total rebellion number \(\mathit{trb}(G)\) is the minimum cardinality of any total rebellion set in \(G\). A total rebellion set with cardinality \(\mathit{trb}(G)\) is denoted by \(\mathit{trb}(G)\)-set. In this paper, we defined the total rebellion number for simple graphs. Also, we determined its tight bounds for some standard graph and characterize this parameters.