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 $k$ 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 $n\geq (k+1)(2k+3)$

When an optimal dominating set with given constraints exists

International audienceA dominating set is a set S of vertices in a graph such that every vertex not in S is adjacent to a vertex in S. In this paper, we consider the set of all optimal (i.e. smallest) dominating sets S, and ask of the existence of at least one such set S with given constraints. The constraints say that the number of neighbors in S of a vertex inside S must be in a given set ρ, and the number of neighbors of a vertex outside S must be in a given set σ. For example, if ρ is[1, k], and σ is the nonnegative integers, this corresponds to “ [1, k]-domination.”First, we consider the complexity of recognizing whether an optimal dominating set with given constraints exists or not. We show via two different reductions that this problem is NP-hard for certain given constraints. This, in particular, answers a question of [M. Chellali et al., [1-2]-dominating sets in graphs, Discrete Applied Mathematics 161 (2013) 2885–2893] regarding the constraint that the number of neighbors in the set be upper-bounded by 2. We also consider the corresponding question regarding “total” dominating sets.Next, we consider some well-structured classes of graphs, including permutation and interval graphs (and their subfamilies), and determine exactly the smallest k such that for all graphs in that family an optimal dominating set exists where every vertex is dominated at most k times. We also consider the problem for trees (with implications for chordal and comparability graphs) and graphs with bounded “asteroidal number”