Independent coalition in graphs: existence and characterization

An independent coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$ neither of which is an independent dominating set but whose union $V_1 \cup V_2$ is an independent dominating set. An independent coalition partition, abbreviated, $ic$-partition, in a graph $G$ is a vertex partition $\pi= \lbrace V_1,V_2,\dots ,V_k \rbrace$ such that each set $V_i$ of $\pi$ either is a singleton dominating set, or is not an independent dominating set but forms an independent coalition with another set $V_j \in \pi$. The maximum number of classes of an $ic$-partition of $G$ is the independent coalition number of $G$, denoted by $IC(G)$. In this paper we study the concept of $ic$-partition. In particular, we discuss the possibility of the existence of $ic$-partitions in graphs and introduce a family of graphs for which no $ic$-partition exists. We also determine the independent coalition number of some classes of graphs and investigate graphs $G$ of order $n$ with $IC(G)\in\{1,2,3,4,n\}$ and the trees $T$ of order $n$ with $IC(T)=n-1$.