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 $Φ$ of $V(G)$ such that each set in $Φ$ either consists of only a singe vertex with the degree $|V(G)|-1$, or forms a connected coalition of $G$ with another set in $Φ$. 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.

Figures (1)

• Figure 1: (a) The family $\mathcal{G}$. (b) $C_{4}+e$.

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10
• Lemma 2.2