For the terminology and notations not defined here, we adopt those in Bondy and Murty [1] and Xu [2] and consider simple graphs only. Let G = (V,E) be a graph with vertex set V = V (G) and edge set E = E(G). For any vertex v ∈ V , NG(v) denotes the open neighborhood of v in G and NG[v] = NG(v) ∪ {v} the closed one. dG(v) = |NG(v)| is called the degree of v in G, ∆ and δ denote the maximum degre...

We prove for the contact process on Z, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate λ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounde...

The domination number γ(G) of a graph G is the minimum cardinality of a subset D of V (G) with the property that each vertex of V (G) − D is adjacent to at least one vertex of D. For a graph G with n vertices we define ǫ(G) to be the number of leaves in G minus the number of stems in G, and we define the leaf density ζ(G) to equal ǫ(G)/n. We prove that for any graph G with no isolated vertex, γ...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

Dunbar et al. (1998) in Ref. [3] introduced the OI~/TLLS ck~r~in~rtio~ ~IWH/W ;,-(G) of a graph G and two open problems. In this paper, we show that for every negative integer k and positive integer m>,3. there exists a graph G with gn-th nl and ;‘(G) <k which is a positive answer for the open problem 2 in Ref. [3].

We consider finite graphs G with vertex set V (G). A subset D ⊆ V (G) is a dominating set of the graph G, if every vertex v ∈ V (G) − D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality among the dominating sets of G. In this note, we characterize the trees T with an even number of vertices such that γ(T ) = |V (T )| − 2

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...