In this paper we extend the notion of weak degree domination in graphs to hypergraphs and find relationships among the domination number, the weak edge-degree domination number, the independent domination number and the independence number of a given hypergraph.

Let G = (V,E) be a graph and f be a function f:V → 0,1,2. A vertex u with f(u) = 0 is said to be undefended with respect to f, if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f’: V → 0,1,2 defined by f’(u) = 1, f’(v) = f(v)-1 and f’(w) = f(w) if w ∈ V-u,v, has no undefended vertex. The weight of f is $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$. The weak Roman domination number,...

A dominating set $D\subseteq V\left(G\right)$ is a weakly connected dominating set in $G$ if the subgraph $G{\left[D\right]}_w=(N_G\left[D\right],E_w)$ weakly induced by $D$ is connected, where $E_w$ is the set of all edges having at least one vertex in $D$. Weakly connected domination number${\gamma }_w\left(G\right)$ of a graph $G$ is the minimum cardinality among all weakly connected dominating sets in $G$. A graph $G$ is said to be weakly connected domination stable or just ${\gamma }_w$-stable if ${\gamma }_w\left(G\right)={\gamma }_w(G+e)$ for every edge $e$ belonging to the complement $\overline{G}$ of $G.$ We provide a constructive characterization of weakly connected domination...

A set D of vertices in a graph G = (V,E) is a weakly connected dominating set of G if D is dominating in G and the subgraph weakly induced by D is connected. The weakly connected domination number of G is the minimum cardinality of a weakly connected dominating set of G. The weakly connected domination subdivision number of a connected graph G is the minimum number of edges that must be subdivided (where each egde can be subdivided at most once) in order to increase the weakly connected domination...