Secure coalitions in graphs

A secure coalition in a graph $G$ consists of two disjoint vertex sets $V_1$ and $V_2$, neither of which is a secure dominating set, but whose union $V_1 \cup V_2$ forms a secure dominating set. A secure coalition partition ($sec$-partition) of $G$ is a vertex partition $\pi = \{V_1, V_2, \dots, V_k\}$ where each set $V_i$ is either a secure dominating set consisting of a single vertex of degree $n-1$, or a set that is not a secure dominating set but forms a secure coalition with some other set $V_j \in \pi$. The maximum cardinality of a secure coalition partition of $G$ is called the secure coalition number of $G$, denoted $SEC(G)$. For every $sec$-partition $\pi$ of a graph $G$, we associate a graph called the secure coalition graph of $G$ with respect to $\pi$, denoted $SCG(G,\pi)$, where the vertices of $SCG(G,\pi)$ correspond to the sets $V_1, V_2, \dots, V_k$ of $\pi$, and two vertices are adjacent in $SCG(G,\pi)$ if and only if their corresponding sets in $\pi$ form a secure coalition in $G$. In this study, we prove that every graph admits a $sec$-partition. Further, we characterize the graphs $G$ with $SEC(G) \in \{1,2,n\}$ and all trees $T$ with $SEC(T) = n-1$. Finally, we show that every graph $G$ without isolated vertices is a secure coalition graph.