This paper focuses on the average order of dominating sets a graph. We find extremal graphs for maximum and minimum value over all n vertices, while trees we prove that star minimizes sets. in without isolated vertices is at most 3n/4, but provide evidence actual upper bound 2n/3. Finally, show normalized average, dense [1/2,1], tends to 12 almost graphs.

The concept of triple connected graphs with real life application was introduced in [7] by considering the existence of a path containing any three vertices of a graph G. In this paper, we introduce a new domination parameter, called Smarandachely triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be Smarandachely triple connected dominating set, i...

A dominating set S of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LDcodes and the cardinality of an LD-code is the location-domination number, λ(G). An LD-set S of a graph G is global if it is an LD-set for both G and its complem...

A dominating set S of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LDcodes and the cardinality of an LD-code is the location-domination number, λ(G). An LD-set S of a graph G is global if it is an LD-set for both G and its complem...

A two-valued function f defined on the vertices of a graph G = (V,E), f : V → {−1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f(N(v)) ≥ 1, where N(v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f(V ) = ∑ f(v), over all vertices v ∈ V . The total ...

Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DR...

An exponential dominating set of graph G = (V,E) is a subset D ⊆ V such that ∑ w∈D( 1 2 )d(v,w)−1 ≥ 1 for every v ∈ V, where d(v, w) is the distance between vertices v and w. The exponential domination number, γe(G), is the smallest cardinality of an exponential dominating set. Lower and upper bounds for γe(Cm × Cn) are determined and it is shown that limm,n→∞ γe(Cm×Cn) mn ≤ 1 13 . Two connecti...

A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of some results on homogeneous sets we present a linear time algorithm computing the vertex sets of the ...

An edge dominating set of a graph G = (V,E) is a subset M ⊆ E of edges in the graph such that each edge in E −M is incident with at least one edge in M . In an instance of the parameterized edge dominating set problem we are given a graph G = (V,E) and an integer k and we are asked to decide whether G has an edge dominating set of size at most k. In this paper we show that the parameterized edg...