Bipartite graphs with close domination and k-domination numbers

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number $\gamma_k(G)$ is the minimum cardinality of a $k$-dominating set in $G$. For any graph $G$, we know that $\gamma_k(G) \geq \gamma(G)+k-2$ where $\Delta(G)\geq k\geq 2$ and this bound is sharp for every $k\geq 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\geq 3$ and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when $k=3$. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general

Equality of domination and transversal numbers in hypergraphs

A subset <i>S</i> of the vertex set of a hypergraph ℋ is called a dominating set of ℋ if for every vertex <i>v</i> not in <i>S</i> there exists <i>u ∈ S</i> such that <i>u</i> and <i>v</i> are contained in an edge in ℋ. The minimum cardinality of a dominating set in ℋ is called the domination number of ℋ and is denoted by γ(ℋ). A transversal of a hypergraph ℋ is defined to be a subset <i>T</i> of the vertex set such that <i>T ⋂ E ≠ Ø</i> for every edge <i>E</i> of ℋ. The transversal number of ℋ, denoted by <i>t</i>.(ℋ), is the minimum number of vertices in a transversal. A hypergraph is of rank <i>k</i> if each of its edges contains at most <i>k</i> vertices. The inequality <i>t</i>(ℋ) = γ(ℋ) is valid for every hypergraph ℋ without isolated vertices. In this paper, we investigate the hypergraphs satisfying <i>t</i>(ℋ) = γ(ℋ), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class ϴ ^p₂. Structurally we focus our attention on hypergraphs in which each subhypergraph ℋ¹ without isolated vertices fulfills the equality <i>t</i>(ℋ¹) = (ℋ¹). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer <i>k</i>, there are only a finite number of forbidden subhypergraphs of rank <i>k</i>, and each of them has domination number at most <i>k</i>