On the connected coalition number

Abstract

For a graph $G=(V,E)$, a pair of vertex disjoint sets $A_1$ and $A_2$ form a connected coalition of $G$, if $A_1\cup A_2$ is a connected dominating set, but neither $A_1$ nor $A_2$ is a connected dominating set. A connected coalition partition of $G$ is a partition $\Phi$ of $V(G)$ such that each set in $\Phi$ either consists of only a singe vertex with the degree $|V(G)|-1$, or forms a connected coalition of $G$ with another set in $\Phi$. The connected coalition number of $G$, denoted by $CC(G)$, is the largest possible size of a connected coalition partition of $G$. In this paper, we characterize graphs that satisfy $CC(G)=2$. Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.