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...

Let G be a graph with no isolated vertex and f : V ( ) → {0, 1, 2} function. i = { x ∈ } for every . We say that is total Roman dominating function on if in 0 adjacent to at least one 2 the subgraph induced by 1 ∪ has vertex. The weight of ω ∑ v minimum among all functions domination number , denoted γ t R It known general problem computing NP-hard. In this paper, we show H nontrivial graph, th...

‎Let $G$ be a finite and simple graph with vertex set $V(G)$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph $G$ is a function $f:V(G)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎where $N(v)$ is the open neighborhood of $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u...

Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f...