We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, dominating function $f$ is called (RDRD function) if subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices. The number number) $\gamma_{rdR}(G)$ minimum weight $\sum_{v\in V(G)}f(v)$ taken over all RDRD functions $G$. first prove that problem computing $\gamma_{rdR}$...

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is double Roman dominating of if for each v ∈ with f(v) = 0, has two adjacent vertices u and w which f(u) f(w) 2 or an vertex 3, to either 3. The minimum weight ωG(f) P v∈V the domination number G. In this paper, we continue study introduced studied by R.A. Beeler et al. in [2]. First, characterize some numbers small values terms 2...

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

Define a Roman dominating function (RDF) of a graph G to be a function f : V (G) → {0, 1, 2} such that every u with f(u) = 0 has a neighbor v with f(v) = 2. The weight of f , w(f), is ∑ v∈V (G) f(v). The Roman domination number of G, γR(G), is the minimum weight of an RDF of G. It is easy to see that γ(G) ≤ γR(G) ≤ 2γ(G), where γ(G) is the domination number of G. In this paper, we determine pro...

For a graph property P and a graph G, a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. A P-Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the set of all vertices with label 1 or 2 is a P-set. The P-Roman domination number γPR(G) of G is the minimum of Σv∈V (...

Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any integers k1, k2, k3, k4, k5 there exists a cubic graph G satisfying the following conditions: γ(G)−ir(G) ≥ k1, i(G)−...

‎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 be a graph. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we would like to characterize the cubic graphs with total domatic number at least two.

We provide two algorithms counting the number of minimum Roman dominating functions of a graph on n vertices in (1.5673) n time and polynomial space. We also show that the time complexity can be reduced to (1.5014) n if exponential space is used. Our result is obtained by transforming the Roman domination problem into other combinatorial problems on graphs for which exact algorithms already exist.