Paired coalition in graphs (2402.10842v4)

Abstract: \noindent A paired coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a paired dominating set but whose union $V_1 \cup V_2$ is a paired dominating set. A paired coalition partition (abbreviated $pc$-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 $\pi$ is not a paired dominating set but forms a paired coalition with another set $V_j \in \pi$. The paired coalition graph $PCG(G,\pi) $ of the graph $G$ and the $pc$-partition $\pi$ of $G$, is the graph whose vertices correspond one-to-one with the sets of $\pi$, and two vertices $V_i$ and $V_j$ are adjacent in $PCG(G,\pi) $ if and only if their corresponding sets $V_i$ and $V_j$ form a paired coalition in $G$. In this paper, we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs $G$ with $PC(G)=n$ and unicyclic graphs $G$ with $PC(G)=n-2$.