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...

In a graph, a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number γ×2(G) is the minimum cardinality of a double dominating set of G. A graph G without isolated vertices is called edge removal critical with respect to double domination, or just γ×2-criti...

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...

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum numbe...

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 finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2...