Abstract:

Let $G$ be a graph of minimum degree at least two. A set $D$ of vertices of a graph $G$ with the vertex set $V$ is a double total dominating set of $G$, if every vertex $v$ has at least two neighbors in $D$. A double total coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a double total dominating set but their union $V_{1}\cup V_{2}$  is a double total dominating set. A double total coalition partition of a graph $G$ is a partition $\Theta=\{V_1, V_2,..., V_k \}$ of $V$ such that no subset of $\Theta$ is a double total dominating set of $G$, but for every set $V_i \in \Theta$, there exists a set $V_j \in \Theta$ such that $V_i$ and $V_j$ form a double total coalition. In this paper we initiate the study of the double total coalition by setting some basic results, giving exact values and bounds for the double total coalition number.