Total Roman Domination Number Research Articles

Total Roman domination in the lexicographic product of graphs

A total Roman dominating function of a graph G=(V,E) is a function f:V(G)→{0,1,2} such that for every vertex v with f(v)=0 there exists a vertex u adjacent to v with f(u)=2, and such that the subgraph induced by the set of vertices labeled one or two has no isolated vertices. The total Roman domination number of G is the minimum value of the sums ∑v∈Vf(v) over all total Roman dominating functions f of G. In this paper we study the total Roman domination number of the lexicographic product of graphs.

Twin signed total Roman domination numbers in digraphs

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text] and arc set [Formula: see text]. A twin signed total Roman dominating function (TSTRDF) on the digraph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) [Formula: see text] and [Formula: see text] for each [Formula: see text], where [Formula: see text] (respectively [Formula: see text]) consists of all in-neighbors (respectively out-neighbors) of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an in-neighbor [Formula: see text] and an out-neighbor [Formula: see text] with [Formula: see text]. The weight of an TSTRDF [Formula: see text] is [Formula: see text]. The twin signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an TSTRDF on [Formula: see text]. In this paper, we initiate the study of twin signed total Roman domination in digraphs and we present some sharp bounds on [Formula: see text]. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

The signed total Roman domatic number of a digraph

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text]. A signed total Roman dominating function (STRDF) on a digraph [Formula: see text] is a function [Formula: see text] such that (i) [Formula: see text] for every [Formula: see text], where [Formula: see text] consists of all inner neighbors of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an inner neighbor [Formula: see text] for which [Formula: see text]. The weight of an STRDF [Formula: see text] is [Formula: see text]. The signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an STRDF on [Formula: see text]. A set [Formula: see text] of distinct STRDFs on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a signed total Roman dominating family (STRD family) (of functions) on [Formula: see text]. The maximum number of functions in an STRD family on [Formula: see text] is the signed total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of signed total Roman domatic number in digraphs and we present some sharp bounds for [Formula: see text]. In addition, we determine the signed total Roman domatic number of some classes of digraphs.

Computational Complexity of Outer-Independent Total and Total Roman Domination Numbers in Trees

An outer-independent total dominating set (OITDS) 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 $V(G)\setminus D$ is independent. The outer-independent total domination number of a graph $G$ , denoted by $\gamma _{oit}(G)$ , is the minimum cardinality of an OITDS of $G$ . An outer-independent total Roman dominating function (OITRDF) on a graph $G$ is a function $f: V(G) \rightarrow \{0, 1, 2\}$ satisfying the conditions that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$ , every vertex $x$ with $f(x)\geq 1$ is adjacent to at least one vertex $y$ with $f(y)\geq 1$ , and any two different vertices $a,b$ with $f(a)=f(b)=0$ are not adjacent. The minimum weight $\omega (f) =\sum _{v\in V(G)}f(v)$ of any OITRDF $f$ for $G$ is the outer-independent total Roman domination number of $G$ , denoted by $\gamma _{oitR}(G)$ . A graph $G$ is called an outer-independent total Roman graph (OIT-Roman graph) if $\gamma _{oitR}(G)=2\gamma _{oit}(G)$ . In this paper, we propose dynamic programming algorithms to compute the outer-independent total domination number and the outer-independent total Roman domination number of a tree, respectively. Moreover, we characterize all OIT-Roman graphs and give a linear time algorithm for recognizing an OIT-Roman graph.

Global total Roman domination in graphs

A total Roman dominating function (TRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions (i) every vertex [Formula: see text] for which [Formula: see text] is adjacent at least one vertex [Formula: see text] for which [Formula: see text] and (ii) the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. The weight of a TRDF is the sum of its function values over all vertices. A total Roman dominating function [Formula: see text] is called a global total Roman dominating function (GTRDF) if [Formula: see text] is also a TRDF of the complement [Formula: see text] of [Formula: see text]. The global total Roman domination number of [Formula: see text] is the minimum weight of a GTRDF on [Formula: see text]. In this paper, we initiate the study of global total Roman domination number and investigate its basic properties. In particular, we relate the global total Roman domination and the total Roman domination and the global Roman domination number.

On the signed total Roman domination and domatic numbers of graphs

A signed total Roman dominating function on a graph G is a function f:V(G)⟶{−1,1,2} such that ∑u∈N(v)f(u)≥1 for every vertex v∈V(G), where N(v) is the neighborhood of v, and every vertex u∈V(G) for which f(u)=−1 is adjacent to at least one vertex w for which f(w)=2. The signed total Roman domination number of a graph G, denoted by γstR(G), equals the minimum weight of a signed total Roman dominating function. A set {f1,f2,…,fd} of distinct signed total Roman dominating functions on G with the property that ∑i=1dfi(v)≤1 for each v∈V(G), is called a signed total Roman dominating family (of functions) on G. The maximum number of functions in a signed total Roman dominating family on G is the signed total Roman domatic number of G, denoted by dstR(G). In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for γstR(G) and dstR(G). In addition, we determine the total signed Roman domatic number of some graphs.

Total Roman domination in graphs

A Roman dominating function on a graph G is a function f:V(G) → {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 a Roman dominating function f is the sum, ΣuV(G) f(u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.

Signed total Roman domination in graphs

Let $$G$$G be a finite and simple graph with vertex set $$V(G)$$V(G). A signed total Roman dominating function (STRDF) on a graph $$G$$G is a function $$f:V(G)\rightarrow \{-1,1,2\}$$f:V(G)ź{-1,1,2} satisfying the conditions that (i) $$\sum _{x\in N(v)}f(x)\ge 1$$źxźN(v)f(x)ź1 for each vertex $$v\in V(G)$$vźV(G), where $$N(v)$$N(v) is the neighborhood of $$v$$v, and (ii) every vertex $$u$$u for which $$f(u)=-1$$f(u)=-1 is adjacent to at least one vertex $$v$$v for which $$f(v)=2$$f(v)=2. The weight of an SRTDF $$f$$f is $$\sum _{v\in V(G)}f(v)$$źvźV(G)f(v). The signed total Roman domination number $$\gamma _{stR}(G)$$źstR(G) of $$G$$G is the minimum weight of an STRDF on $$G$$G. In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on $$\gamma _{stR}(G)$$źstR(G). In addition, we determine the signed total Roman domination number of some classes of graphs.