In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of...

Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. Introduced by Sumner and Moorer [23], a graph G is domination perfect if γ(H) = ι(H) for every induced subgraph H ⊆ G. In 1991, Zverovich and Zverovich [26] proposed a characterization of domination perfect graphs in terms of forbidden induced subgraphs. Fulman [15] noticed that this characterizat...

In this note the split domination number of the Cartesian product of two paths is considered. Our results are related to [2] where the domination number of Pm¤Pn was studied. The split domination number of P2¤Pn is calculated, and we give good estimates for the split domination number of Pm¤Pn expressed in terms of its domination number.

We provide a simple constructive characterization for trees with equal domination and independent domination numbers, and for trees with equal domination and total domination numbers. We also consider a general framework for constructive characterizations for other equality problems.

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs total domination critical or just γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We study an open problem of ...

The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt(G) of a graph G and we show that for any connected graph G of order at least two, msdγt(G) ≤ 3. We show that...

In this paper, we propose a new network reliability measure for some particular kind of service networks, which we refer to as domination reliability. We relate this new reliability measure to the domination polynomial of a graph and the coverage probability of a hypergraph. We derive explicit and recursive formulæ for domination reliability and its associated domination reliability polynomial,...

for a graph $g$ let $gamma (g)$ be its domination number. we define a graph g to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ed}$ graph) if $g$ has no efficient dominating set (eds) but every graph formed by removing a single vertex from $g$ has at least one eds, and (ii) a hypo-unique domination graph (a hypo-$mathcal{ud}$ graph) if $g$ has at least two minimum dominating sets...

the broadcast domination number of the cartesian product of two cycles is completely determined.