‎In this paper‎, ‎we investigate domination number‎, ‎$gamma$‎, ‎as well‎ ‎as signed domination number‎, ‎$gamma_{_S}$‎, ‎of all cubic Cayley‎ ‎graphs of cyclic and quaternion groups‎. ‎In addition‎, ‎we show that‎ ‎the domination and signed domination numbers of cubic graphs depend‎ on each other‎.

A set S of vertices of a graphG = (V,E) is a dominating set if every vertex of V (G)\S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Velammal ...

We study the parameterized complexity of domination-type problems. (σ, ρ)-domination is a general and unifying framework introduced by Telle: a set D of vertices of a graph G is (σ, ρ)dominating if for any v ∈ D, |N(v) ∩ D| ∈ σ and for any v / ∈ D, |N(v) ∩ D| ∈ ρ. We mainly show that for any σ and ρ the problem of (σ, ρ)-domination is W[2] when parameterized by the size of the dominating set. T...

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...

Let Y be a subset of real numbers. A Y dominating function of a graph G = (V, E) is a function f : V → Y such that u∈NG[v] f(u) ≥ 1 for all vertices v ∈ V , where NG[v] = {v} ∪ {u|(u, v) ∈ E}. Let f(S) = u∈S f(u) for any subset S of V and let f(V ) be the weight of f . The Y -domination problem is to find a Y -dominating function of minimum weight for a graph. In this paper, we study the variat...

Decision making in the presence of multiple and conflicting objectives requires preference from the decision maker. The decision maker’s preferences give rise to a domination structure. Till now, most of the research has been focussed on the standard domination structure based on the Pareto-domination principle. However, various real world applications like medical image registration, financial...

A paired dominating set $P$ is a with the additional property that has perfect matching. While maximum cardainality of minimal in graph $G$ called upper domination number $G$, denoted by $\Gamma(G)$, cardinality $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any without isolated vertices. We focus on graphs satisfying equality $\Gamma_{pr}(G)= 2\Gam...

We consider an uncertain version of the scheduling problem to sequence set of jobs J on a single machine with minimizing the weighted total flow time, provided that processing time of a job can take on any real value from the given closed interval. It is assumed that job processing time is unknown random variable before the actual occurrence of this time, where probability distribution of such ...

A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let c(G) denote the size of any smallest connected dominating set in G. A graph G is k-connected-critical if c(G)= k, but if any edge e ∈ E(Ḡ) is added to G, then c(G+ e) k − 1. This is a variation on the earlier concept of criticality of edge addition with respect to ordinary domination where a graph G ...