On the number of edges in graphs with a given weakly connected domination number

The asymptotic number of labelled weakly-connected digraphs with a given number of vertices and edges

Edward A. Benderl Depariment of Mathel1wtics, University of California, San Diego, La Jolla, CA 92093, USA E. Rodney Canfieldl Department of Computer Science, University of Georgia, Athens, GA 30602, USA Brendan D. McKay2 Computer Science Department, Australian National University, GPO Box 4, ACT 2601., Australia In a recent paper, we determined the asymptotic number of labelled connected graph...

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...

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...

The edge domination number of connected graphs

A subset X of edges in a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. Let m,n and k be positive integers with n − 1 ≤ m ≤ (n 2 ) , G(m,n) be the set of all non-isomorphic connected graphs of order n and size m, and G(m,n; k) = {G ∈ G(m...

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