Introduction to total coalitions in graphs

Abstract

Let $G$ be a graph with vertex set $V$. Two disjoint sets $V_1, V_2\subseteq V$ are called a total coalition in $G$, if neither $V_1$ and $V_2$ is a total dominating set of $G$ but $V_1\cup V_2$ is a total dominating set. A total coalition partition of $G$ is a vertex partition $\pi=\{V_1,V_2,\ldots, V_k\}$ such that no set of $\pi$ is a total dominating set but each set $V_i\in \pi$ forms a total coalition with another set $V_j\in \pi$. The maximum cardinality of a total coalition partition of $G$ is called the total coalition number of $G$, denoted by $TC(G)$. In this paper, we initiate the study of the total coalition in graphs and its properties.