On the distance domination number of bipartite graphs

TL;DR

This paper establishes upper bounds on the k-distance domination number in connected bipartite graphs, improving existing theoretical results and contributing to the understanding of domination parameters in graph theory.

Contribution

It provides new upper bounds for the k-distance domination number in bipartite graphs, refining previous results in the literature.

Findings

Derived tighter upper bounds for the k-distance domination number.

Improved upon previous theorems by Tian and Xu.

Enhanced theoretical understanding of domination in bipartite graphs.

Abstract

A subset $D\subseteq V(G)$ is called a $k$-distance dominating set of $G$ if every vertex in $V(G)\setminus D$ is within distance $k$ from some vertex of $D$. The minimum cardinality among all $k$-distance dominating sets of $G$ is called the $k$-distance domination number of $G$. In this note we give upper bound on the $k$-distance domination number of a connected bipartite graph and improve some results have been given like Theorem 2.1 and 2,7 in [Tian and Xu, A note on distance domination of graphs, Australian Journal of Combinatorics, 43 (2009), 181-190].