Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

The concept of inverse domination was introduced by Kulli V.R. and Sigarakanti S.C. [9] . Let D be a  set of G. A dominating set D1  VD is called an inverse dominating set of G with respect to D. The inverse domination number   (G) is the order of a smallest inverse dominating set. Motivated by this definition we define another parameter as follows. Let D be a maximum independent set in G. ...

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a 2-dominating set of G if S dominates every vertex of V (G) \ S at least twice. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set of G. The 2-domination subdivision number sdγ2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in ...

For any $k in mathbb{N}$, the $k$-subdivision of graph $G$ is a simple graph $G^{frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$, denoted by ...

A set S of vertices in a graph G is a 2-dominating set if every vertex of G not in S is adjacent to at least two vertices in S, and S is a 2-independent set if every vertex in S is adjacent to at most one vertex of S. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set in G, and the 2-independence number α2(G) is the maximum cardinality of a 2-independent set in G. Ch...

Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V − S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ2(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). Fink and Jaco...

In a graph, vertex dominates itself and its neighbors. A subset S of vertices graph G is double dominating set if every at least twice. The domination number γ ×2 ( ) the minimum cardinality . this paper, we prove that maximal outerplanar order n bounded above by + k /2, where pairs consecutive degree two with distance 3 on outer cycle. We also ×2( ≤ 5 /8 for Hamiltonian planar ≥ 7.