Some results on characterizing the edges of connected graphs with a given domination number

The Asymptotic Number of Labeled Connected Graphs with a Given Number of Vertices and Edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q 5 N = ( ) edges. Let x = q/n and k = q n. We determine functions w k 1 , a(x ) and cp(x) such that c (n , q) w k ( z ) e n r p ( x ) e o ( x ) uniformly for all n and q 2 n. If Q > O is fixed, n+= and 4q > ( 1 + ~ ) n log n, this formula simplifies to c(n, q) ( t ) exp(-ne-zq’n). On the other hand, if k = o(n”’), this f...

Some Results on the Maximal 2-Rainbow Domination Number in Graphs

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...

On the super domination number of graphs

The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ such that $N(v)cap overline{D}={u}$. The super domination number of $G$ is the minimum car...

Preliminaries on the Asymptotic Number of H-free Labeled Connected Graphs with a given Number of Vertices and Edges

On the Domination Number of Some Graphs

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of graph G, if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. It is well known that if e ∈ E(G), then γ(G− e)−1 ≤ γ(G) ≤ γ(G− e). In this paper, as an application of this inequality, we obtain the domination number of some certain graph...