The maximum number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.7159. This upper bound might not be tight, since no examples of graphs with 1.5705 or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the maximum number of minimal dominating sets in graphs on n vertices. In some cases, we ...

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (G), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = ∑ v∈V f(v). The total {k}-dominati...

A subsetD of the vertex set of a graph G, is a dominating set if every vertex in V −D is adjacent to at least one vertex inD. The domination number γ G is the minimum cardinality of a dominating set of G. A subset of V −D, which is also a dominating set of G is called an inverse dominating set of G with respect toD. The inverse domination number γ ′ G is the minimum cardinality of the inverse d...

date palm (phoenix dactylifera l.) with its delicious and nutritious bearing fruit (dates) is an important agricultural core in iran. to extend the shelf life of date palm fruit, some such methods of preservation as drying are suggested and drying influences physicochemical and quality characteristics of products, thus modeling of drying kinetics is one of the tools for process control. in this...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γt. The maximal size of an inclusionwise minimal total dominating set, the upper total domination num...

A set D of a graph G = (V,E) is a dominating set if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. A dominating set D is called a complementary tree dominating set if the induced sub graph < V −D > is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domin...

In this paper we consider the CAPACITATED DOMINATING SET problem — a generalisation of the DOMINATING SET problem where each vertex v is additionally equipped with a number c(v), which is the number of other vertices this vertex can dominate. We provide an algorithm that solves CAPACITATED DOMINATING SET exactly in O(1.89) time and polynomial space. Despite the fact that the CAPACITATED DOMINAT...

Let G be a connected simple graph. A restrained dominating set S of the vertex set of G, V (G) is a secure restrained dominating set of G if for each u ∈ V (G) \ S, there exists v ∈ S such that uv ∈ E(G) and the set (S \ {v}) ∪ {u} is a restrained dominating set of G. The minimum cardinality of a secure restrained dominating set of G, denoted by γsr(G), is called the secure restrained dominatio...