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 (...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

Domination theory is a well-established topic in graph theory, as well one of the most active research areas. Interest this area partly explained by its diversity applications to real-world problems, such facility location computer and social networks, monitoring communication, coding algorithm design, among others. In last two decades, functions defined on graphs have attracted attention sever...

In this paper we study the signed Roman dominationnumber of the join of graphs. Specially, we determine it for thejoin of cycles, wheels, fans and friendship graphs.