Let G be an undirected graph with vertex and edge sets V (G) E(G), respectively. A set S ⊆ is called a hop independent dominating of if both G. The minimum cardinality G, denoted by γhih(G), the domination number In this paper, we show that lies between independence We characterize these types in shadow graph, join, corona, lexicographic product two graphs. Moreover, either exact values or boun...

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2dominating set if every vertex of V (...

For a graph G = (V,E), a subset D ⊆ V (G) is a total dominating set if every vertex of G has a neighbor in D. The total domination number of G is the minimum cardinality of a total dominating set of G. A subset D ⊆ V (G) is a 2-dominating set of G if every vertex of V (G) \ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V (G) \ D...

In this paper, we consider various types of domination vertex critical graphs, including total domination vertex critical graphs and independent domination vertex critical graphs and connected domination vertex critical graphs. We provide upper bounds on the diameter of them, two of which are sharp. MSC (2010): 05C12, 05C69

An outer-independent double Italian dominating function (OIDIDF)on a graph $G$ with vertex set $V(G)$ is a function$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,and the set $ {uin V(G)|f(u)=0}$ is independent. The weight ofan OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. Theminimum weight of an OIDIDF on a graph $G$ is cal...

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

4 In this paper, we study the domination number, the global dom5 ination number, the cographic domination number, the global co6 graphic domination number and the independent domination number 7 of all the graph products which are non-complete extended p-sums 8 (NEPS) of two graphs. 9

Given a semigraph, we can construct graphs Sa, Sca, Se and S1e. In the same pattern, we construct bipartite graphs CA(S), A(S), VE(S), CA(S) and A(S). We find the equality of domination parameters in the bipartite graphs constructed with the domination and total domination parameters of the graphs Sa and Sca. We introduce the domination and independence parameters for the bipartite semigraph. W...

The domination number γ(G), the independent domination number ι(G), the connected domination number γc(G), and the paired domination number γp(G) of a graph G (without isolated vertices, if necessary) are related by the simple inequalities γ(G) ≤ ι(G), γ(G) ≤ γc(G), and γ(G) ≤ γp(G). Very little is known about the graphs that satisfy one of these inequalities with equality. I.E. Zverovich and V...