In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...

In this paper, we introduce the closed domination in graphs. Some interesting relationships are known between domination and closed domination and between closed domination and the independent domination. It is also shown that any triple m, k and n of positive integers with 3 ≤ m ≤ k ≤ n are realizable as the domination number, closed domination number and independent domination number, respect...

Two new domination parameters for a connected graph G: the weakly convex domination number of G and the convex domination number of G are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number.

In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G...

In this paper, we introduce the concept of k-power domination which is a common generalization of domination and power domination. We extend several known results for power domination to k-power domination. Concerning the complexity of the k-power domination problem, we first show that deciding whether a graph admits a k-power dominating set of size at most t is NP-complete for chordal graphs a...