A function f : V (G) → {0, 1, 2} is a Roman dominating function for a graph G = (V,E) if for every vertex v with f(v) = 0, there exists a vertex w ∈ N(v) with f(w) = 2. Emperor Constantine had the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armi...

A Roman dominating function of 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. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V → {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. The weight of f is w(f) = ∑ v∈V f(v). The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above...

We initiate the study of total outer-independent domination in graphs. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \ D is independent. The total outer-independent domination number of a graph G is the minimum cardinality of a total outer-independent dominating set of G. First we discuss the ...

A double Roman dominating function of a graph $G$ is $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$ with $f(v)=0$, there exists $u\in N(v)$ $f(u)=3$, or are $u,w\in $f(u)=f(w)=2$, and if $f(v)=1$, then adjacent to assigned at least $2$ under $f$. The domination number $\gamma_{dR}(G)$ minimum weight $f(V(G))=\sum_{v\in V(G)}f(v)$ among all functions $G$. An outer i...

A function f : V (G) → {0, 1, 2} is a Roman dominating function if for every vertex with f(v) = 0, there exists a vertex w ∈ N(v) with f(w) = 2. We introduce two fractional Roman domination parameters, γR ◦ f and γRf , from relaxations of two equivalent integer programming formulations of Roman domination (the former using open neighborhoods and the later using closed neighborhoods in the Roman...

A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V (G)\D has a at least two neighbors in D, and the set V (G) \D is independent. The 2-outer-independent domination number of a graph G, denoted by γ 2 (G), is the minimum cardinality of a 2-outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leav...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...

A total outer-independent dominating set of a graph G = (V (G), E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \D is independent. The total outer-independent domination number of a graph G, denoted by γ t (G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n ≥ 4, with l le...