A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...

The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also pro...

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 ...

For a graph G=(V,E) with V=V(G) and E=E(G), perfect double Italian dominating function is f:V→{0,1,2,3} having the property that 3≤∑u∈NG[v]f(u)≤4, for every vertex v∈G f(v)∈{0,1}. The weight of f sum f(V)=∑v∈V(G)f(v) minimum on G domination number γdIp(G) G. We initiate study functions. check γdIp some standard graphs evaluate γdI such graphs. functions versus Roman are perused. NP-completeness...

Let K n denote the Cartesian product Kn Kn Kn, where Kn is the complete graph on n vertices. We show that the domination number of K n

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...