Total Roman Research Articles

Total Roman domination for proper interval graphs

A function f : V → {0,1,2} is a total Roman dominating function (TRDF) on a graph G =( V,E ) if for every vertex v ∈ V with f ( v ) = 0 there is a vertex u adjacent to v with f ( u ) = 2 and for every vertex v ∈ V with f ( v ) > 0 there exists a vertex u ∈ N G ( v ) with f ( u ) > 0. The weight of a total Roman dominating function f on G is equal to f ( V )=Σ v ∈ V f ( v ). The minimum weight of a total Roman dominating function on G is called the total Roman domination number of G . In this paper, we give an algorithm to compute the total Roman domination number of a given proper interval graph G =( V,E ) in O (| V |) time.

Total Roman Domination Number of Rooted Product Graphs

Let G be a graph with no isolated vertex and f:V(G)→{0,1,2} a function. If f satisfies that every vertex in the set {v∈V(G):f(v)=0} is adjacent to at least one vertex in the set {v∈V(G):f(v)=2}, and if the subgraph induced by the set {v∈V(G):f(v)≥1} has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight ω(f)=∑v∈V(G)f(v) among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.

Dominating the Direct Product of Two Graphs through Total Roman Strategies

Given a graph G without isolated vertices, a total Roman dominating function for G is a function f:V(G)→{0,1,2} such that every vertex u with f(u)=0 is adjacent to a vertex v with f(v)=2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number γtR(G) of G is the smallest possible value of ∑v∈V(G)f(v) among all total Roman dominating functions f. The total Roman domination number of the direct product G×H of the graphs G and H is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between γtR(G×H) and some classical domination parameters for the factors are given. Characterizations of the direct product graphs G×H achieving small values (≤7) for γtR(G×H) are presented, and exact values for γtR(G×H) are deduced, while considering various specific direct product classes.

Further Results on the Total Roman Domination in Graphs

Let G be a graph without isolated vertices. A function f : V ( G ) → { 0 , 1 , 2 } is a total Roman dominating function on G if every vertex v ∈ V ( G ) for which f ( v ) = 0 is adjacent to at least one vertex u ∈ V ( G ) such that f ( u ) = 2 , and if the subgraph induced by the set { v ∈ V ( G ) : f ( v ) ≥ 1 } has no isolated vertices. The total Roman domination number of G, denoted γ t R ( G ) , is the minimum weight ω ( f ) = ∑ v ∈ V ( G ) f ( v ) among all total Roman dominating functions f on G. In this article we obtain new tight lower and upper bounds for γ t R ( G ) which improve the well-known bounds 2 γ ( G ) ≤ γ t R ( G ) ≤ 3 γ ( G ) , where γ ( G ) represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.

Total Roman {3}-domination in Graphs

For a graph G = ( V , E ) with vertex set V = V ( G ) and edge set E = E ( G ) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that ∑ u ∈ N G ( v ) f ( u ) ≥ 3 , if f ( v ) = 0 , and ∑ u ∈ N G ( v ) f ( u ) ≥ 2 , if f ( v ) = 1 for any vertex v ∈ V ( G ) . The weight of a Roman { 3 } -dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } ( G ) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f ( v ) ≠ 0 has a neighbor w with f ( w ) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } ( G ) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs.

Bounds on the signed total Roman 2-domination in graphs

Let [Formula: see text] be an integer and [Formula: see text] be a simple and finite graph with vertex set [Formula: see text]. A signed total Roman [Formula: see text]-dominating function (STR[Formula: see text]DF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] and (ii) [Formula: see text] holds for any vertex [Formula: see text]. The weight of an STR[Formula: see text]DF [Formula: see text] is [Formula: see text] and the minimum weight of an STR[Formula: see text]DF is the signed total Roman [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, we establish some sharp bounds on the signed total Roman 2-domination number.

Total Roman \{2\}-dominating functions in graphs

Total Roman <i>\{2\}</i>-dominating functions in graphs

Algorithmic and complexity aspects of problems related to total Roman domination for graphs

A function $$f:V(G)\rightarrow \{0,1,2\}$$ is a Roman dominating function (RDF) if 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 is the value $$f(V(G))=\sum _{u \in V}f(u)$$. The Roman domination number of a graph G, denoted by $$\gamma _{R}(G)$$, is the minimum weight of a Roman dominating function on G. A connected (respectively, total) Roman dominating function is an RDF f such that the vertices with non-zero labels under f induce a connected graph (respectively, a subgraph with no isolated vertex). The connected (respectively, total) Roman domination number of a graph G, denoted by $$\gamma _{cR}(G)$$ (respectively, $$\gamma _{tR}(G)$$) is the minimum weight of a connected (respectively, total) RDF of G. It this paper we first study the complexity issue of the problems posed in [H. Abdollahzadeh Ahangar, M. A. Henning, V. Samodivkin and I. G. Yero, Total Roman domination in graphs, Appl. Anal. Discret. Math. 10 (2016), 501–517], and show that the problem of deciding whether $$\gamma _{tR}(G)=2\gamma (G)$$, $$\gamma _{tR}(G)=2\gamma _t(G)$$ or $$\gamma _{tR}(G)=3\gamma (G)$$ is NP-hard even when restricted to chordal or bipartite graphs. Then, we give a linear algorithm that decides whether $$\gamma _{tR}(G)=2\gamma (G)$$, $$\gamma _{tR}(G)=2\gamma _t(G)$$ or $$\gamma _{tR}(G)=3\gamma (G)$$, if G is a tree or a unicyclic graph.

Total Roman {2}-domination in graphs

Given a graph G = (V, E), a function f: V → {0, 1, 2} is a total Roman {2}-dominating function if every vertex ʋ ∈ V for which f (ʋ) = 0 satisfies that ∑ u ∈N (ʋ) f (ʋ) ≥ 2, where N (ʋ) represents the open neighborhood of ʋ, and every vertex x ∈ V for which f (x) ≥ 1 is adjacent to at least one vertex y ∈ V such that f (y) ≥ 1. The weight of the function f is defined as ω(f ) = ∑ ʋ ∈V f (ʋ). The total Roman {2}-domination number, denoted by γt {R2}(G), is the minimum weight among all total Roman {2}-dominating functions on G. In this article we introduce the concepts above and begin the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other domination related parameters in graphs. In addition, we prove that the complexity of computing the value γt {R2}(G) is NP-hard, even when restricted to bipartite or chordal graphs.

Total Roman domination in digraphs

Let D be a finite and simple digraph with vertex set V (D). A Roman dominating function (RDF) on a digraph D is a function f : V (D) → {0, 1, 2} satisfying the condition that every vertex v with f (v) = 0 has an in-neighbor u with f (u) = 2. The weight of an RDF f is the value . An RDF f on D with no isolated vertex is called a total Roman dominating function if the subdigraph of D induced by the set {v ∈ V (D): f (v) ≠ 0} has no isolated vertex. The Roman domination number is the minimum weight of a total Roman dominating function on D. In this paper, we initiate the study of the total Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some sharp bounds for the total Roman domination number and we determine the total Roman domination number of some classes of digraphs.

Total Roman domination edge-critical graphs

A total Roman dominating function on a graph $G$ is a function $% f:V(G)\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to some vertex $u$ with $f(u)=2$, and the subgraph of $G$ induced by the set of all vertices $w$ such that $f(w)>0$ has no isolated vertices. The weight of $f$ is $\Sigma _{v\in V(G)}f(v)$. The total Roman domination number $\gamma _{tR}(G)$ is the minimum weight of a total Roman dominating function on $G$. A graph $G$ is $k$-$\gamma _{tR}$-edge-critical if $\gamma _{tR}(G+e)<\gamma _{tR}(G)=k$ for every edge $e\in E(\overline{G})\neq \emptyset $, and $k$-$\gamma _{tR}$-edge-supercritical if it is $k$-$\gamma _{tR}$-edge-critical and $\gamma _{tR}(G+e)=\gamma _{tR}(G)-2$ for every edge $e\in E(\overline{G})\neq \emptyset $. We present some basic results on $\gamma_{tR}$-edge-critical graphs and characterize certain classes of $\gamma _{tR}$-edge-critical graphs. In addition, we show that, when $k$ is small, there is a connection between $k$-$\gamma _{tR}$-edge-critical graphs and graphs which are critical with respect to the domination and total domination numbers.

Total Weak Roman Domination in Graphs

Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) &gt; 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) − 1 and f ′ ( x ) = f ( x ) whenever x ∈ V \ { u , v } , is a total dominating function as well. The weight of a function f is defined to be w ( f ) = ∑ v ∈ V f ( v ) . In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γ t r ( G ) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γ t r ( G ) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.

Total Roman domatic number of a graph

A total Roman dominating function on a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label 0 has a neighbor with label 2 and the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. A set [Formula: see text] of total Roman dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text], is called a total Roman dominating family (of functions) on [Formula: see text]. The maximum number of functions in a total Roman dominating family on [Formula: see text] is the total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total Roman domatic number of some special graphs.

Outer-independent total Roman domination in graphs

Given a graph G with vertex set V, a function f:V→{0,1,2} is an outer-independent total Roman dominating function on G if •every vertex v∈V for which f(v)=0 is adjacent to at least one vertex u∈V such that f(u)=2,•every vertex x∈V for which f(x)≥1 is adjacent to at least one vertex y∈V such that f(y)≥1, and•any two different vertices a,b for which f(a)=f(b)=0 are not adjacent. The minimum weight ω(f)=∑w∈Vf(w) of any outer-independent total Roman dominating function on G is the outer-independent total Roman domination number, γoitR(G), of G. In this article, we introduce the concepts above and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other ones related with domination in graphs. We prove that computing γoitR of a graph G is an NP-hard problem. In addition, we present some closed formulas for γoitR(G) in the cases G represents some special families of graphs.

On the total Roman domination in trees

On the total Roman domination in trees

Total Roman reinforcement in graphs

Total Roman reinforcement in graphs

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.