Let D be a finite and simple digraph with vertex set V (D). A signed total Roman k-dominating function (STRkDF) on D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−(v) f(x) ≥ k for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight o...

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The w...

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...

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 $G$ 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...

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.

Let G = (V, E) be a graph and let S ⊆ V . The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), i...