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

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 fc vertices of D. A [1, k]-dominating set with the minimum number of vertices is called a [formula]-set and the number of its vertices is the [1, k]-domination number [formula] of G. In this short note we show that the decision problem whether [formula] is an NP-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph G of order n satisfying [formula] is given for every integer n ≥ (k + l)(2k + 3).