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arXiv · 1912.03919

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

Narges Ghareghani·Iztok Peterin·Pouyeh Sharifani
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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 $n\geq (k+1)(2k+3)$.

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Narges Ghareghani, Iztok Peterin, Pouyeh Sharifani. 2019-12-09. A note on bipartite graphs whose [1, k]-domination number equal to their number of vertices. https://arxiv.org/abs/1912.03919

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