Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs G and H the upper domination number of the Cartesian product G H is at least the product of their upper domination numbers, in symbols: Γ(G H) ≥ Γ(G)Γ(H). A conjecture posed by Vizing [7] in 1968 claims that Vizing’s conjecture: For any graphs G and H, γ(G H) ≥ γ(G)γ(H), where γ, as usual, denotes the do...

A hypergraph is a pair pV, Eq where V is a finite set and E Ď 2 is called the set of hyper-edges. An output-polynomial algorithm for C Ď 2 is an algorithm that lists without repetitions all the elements of C in time polynomial in the sum of the size of H and the accumulated size of all the elements in C. Whether there exists an output-polynomial algorithm to list all the inclusion-wise minimal ...

A vertex set S is called a power dominating of graph G if every within the system monitored by following collection rules for grid monitoring. The domination number order minimal G. In this paper, we solve splitting and degree graph.

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...

a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the...

Given a graph class G, it is natural to ask whether a given graph has a connected or a total dominating set inducing a graph in G and, if so, what is the minimal size of such a set. We give a sufficient condition on G for the intractability of this problem. This condition is fulfilled by a wide range of graph classes.

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...