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...

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...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...

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 ...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. A set S of vertices in a graph G(V,E) is called a total restrained dominating set if every vertex v ∈ V is adjacent to an element of S and every vertex of V − S is adjacent to a vertex in V − S. The total domination number of a graph G denoted by γt(G) is the minimum card...

The following fundamental result for the domination number γ(G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ(G) ≤ ln(δ + 1) + 1 δ + 1 n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [On double domination in graphs. Discuss. Math. Graph Theory 25 (2005) 29–34...