Paired domination and 2- distance Paired domination of the flower graph $f_{n\times m}$

Abstract

Let $G=(V,E)$ be a graph without an isolated vertex. A set $D\subseteq V(G)$ is a $k$-distance paired domination set of $G$ if $D$ is a $k$-distance dominating set of $G$ and the induced subgraph $⟨D⟩$ has a perfect matching. The minimum cardinality of a $k$-distance paired dominating set for graph $G$ is the $k$-distance paired domination number, denoted by ${\gamma }_p^k(G)$. In this paper, the $k$-distance paired domination of the flower graph $f_{n\times m}$ is discussed. For $m,n\ge 3$, the exact values for paired domination number and $2$-distance paired domination number of flower graph $f_{n\times m}$ are determined

17 pages, 2 figures