A note on bipartite graphs whose [1, k]-domination number equal to their number of vertices

TL;DR

This paper investigates the computational complexity of the [1,k]-domination number in bipartite graphs, proving NP-hardness of the decision problem and providing a construction for graphs where this number equals the total vertices.

Contribution

It establishes NP-hardness of determining the [1,k]-domination number in bipartite graphs and offers a construction method for graphs with maximum [1,k]-domination number.

Findings

Decision problem is NP-hard for bipartite graphs.

Construction of bipartite graphs with maximum [1,k]-domination number.

Applicable for graphs of order n ≥ (k+1)(2k+3).

Abstract

A subset $D$ of the vertex set $V$ of a graph $G$ is called an $[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the minimum number of vertices is called a $\gamma_{[1,k]}$-set and the number of its vertices is the $[1,k]$-domination number $\gamma_{[1,k]}(G)$ of $G$. In this short note we show that the decision problem whether $\gamma_{[1,k]}(G)=n$ is an $NP$-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph $G$ of order $n$ satisfying $\gamma_{[1,k]}(G)=n$ is given for every integer $n\geq (k+1)(2k+3)$.