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

## TL;DR

This paper determines the exact values of the paired domination number and 2-distance paired domination number for flower graphs $f_{n\times m}$ when $m,n \geq 3$, advancing understanding of domination parameters in specific graph classes.

## Contribution

The paper provides the first exact formulas for paired and 2-distance paired domination numbers of flower graphs $f_{n\times m}$ for $m,n \geq 3$, filling a gap in graph domination theory.

## Key findings

- Exact paired domination number for $f_{n\times m}$ with $m,n \geq 3$
- Exact 2-distance paired domination number for $f_{n\times m}$ with $m,n \geq 3$
- Advances understanding of domination parameters in flower graphs

## 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 $\langle D \rangle$ 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 $m,n\geq 3$, the exact values for paired domination number and $2$-distance paired domination number of flower graph $f_{n\times m}$ are determined