Domination numbers and noncover complexes of hypergraphs

Equality of domination and transversal numbers in hypergraphs

The domination number γ(H) and the transversal number τ(H) (also called vertex covering number) of a hypergraph H are defined analogously to those of a graph. A hypergraph is of rank k if each edge contains at most k vertices. The inequality τ(H) ≥ γ(H) is valid for every hypergraph H without isolated vertices. We study the structure of hypergraphs satisfying τ(H) = γ(H), moreover prove that th...

Directed domination in oriented hypergraphs

ErdH{o}s [On Sch"utte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $log (n+1)$ vertices, where $log$ is the logarithm to base $2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $log n - 2 log log n + 1$. This notion of directed domination number has been g...

The Roman domination and domatic numbers of a digraph

A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on ...

Domination game on uniform hypergraphs

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph H by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of H are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay...

Total domination and total domination subdivision numbers