Total Roman Domination Number Research Articles

New bounds on the outer-independent total double Roman domination number

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying (i) if [Formula: see text] then there must be at least two neighbors assigned two under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; and (ii) if [Formula: see text] then [Formula: see text] must be adjacent to a vertex [Formula: see text] such that [Formula: see text]. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on [Formula: see text] if the set of vertices labeled [Formula: see text] induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number [Formula: see text] is the minimum weight of an OITDRDF on [Formula: see text]. In this paper, we establish various bounds on [Formula: see text]. In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results.

Several Roman domination graph invariants on Kneser graphs

This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.

Hop total Roman domination in graphs

In this article, we initiate a study of hop total Roman domination defined as follows: a hop total Roman dominating function (HTRDF) on a graph is a function such that for every vertex u with f(u) = 0 there exists a vertex v at distance 2 from u with f(v) = 2 and the subgraph induced by the vertices assigned non-zero values under f has no isolated vertices. The weight of an HTRDF is the sum of its function values over all vertices, and the hop total Roman domination number equals the minimum weight of an HTRDF on G. We provide several properties on the hop total Roman domination number. More precisely, we show that the decision problem corresponding to the hop total Roman domination problem is NP-complete for bipartite graphs, and we determine the exact value of for paths and cycles. Moreover, we characterize all connected graphs G of order n with Finally, we show that for every tree T of diameter at least 3, where is the hop total domination number.

On the signed strong total Roman domination number of graphs

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominationnumber of special classes of graphs.

Relating the Outer-Independent Total Roman Domination Number with Some Classical Parameters of Graphs

For a given graph G without isolated vertex we consider a function f: V(G) rightarrow {0,1,2}. For every iin {0,1,2}, let V_i={vin V(G):; f(v)=i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V_0 is adjacent to at least one vertex in V_2; (ii) V_0 is an independent set; and (iii) the subgraph induced by V_1cup V_2 has no isolated vertex. The minimum possible weight omega (f)=sum _{vin V(G)}f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

Trees with total Roman domination number equal to Roman domination number plus its domination number: complexity and structural properties

A Roman dominating function on a graph G is a function 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. A total Roman dominating 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 weight of a total Roman dominating function f is the value The total Roman domination number of G, is the minimum weight of a total Roman dominating function in G. In this paper we first study the complexity issue of the problem posed in [A. Cabrera Martinez, S. Cabrera Garcia, A. Carrión Garcia, Further results on the total Roman domination in graphs, Mathematics 8(3) (2020) 349], and then show that the problem of deciding whether is NP-complete. Also, we prove that for any tree T of order if and only if

Closed formulas for the total Roman domination number of lexicographic product graphs

Let G be a graph with no isolated vertex and f : V ( G ) → {0, 1, 2} a function. Let V i = { x ∈ V ( G ) : f ( x ) = i } for every i ∈ {0, 1, 2} . We say that f is a total Roman dominating function on G if every vertex in V 0 is adjacent to at least one vertex in V 2 and the subgraph induced by V 1 ∪ V 2 has no isolated vertex. The weight of f is ω ( f ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight among all total Roman dominating functions on G is the total Roman domination number of G , denoted by γ t R ( G ) . It is known that the general problem of computing γ t R ( G ) is NP-hard. In this paper, we show that if G is a graph with no isolated vertex and H is a nontrivial graph, then the total Roman domination number of the lexicographic product graph G ∘ H is given by γ t R ( G ∘ H ) = 2 γ t ( G ) if γ ( H ) ≥ 2, and γ t R ( G ∘ H ) = ξ ( G ) if γ ( H ) = 1 , where γ ( H ) is the domination number of H , γ t ( G ) is the total domination number of G and ξ ( G ) is a domination parameter defined on G .

On the total Roman domination stability in graphs

A total Roman dominating function on a graph G is a function satisfying the conditions: (i) every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2; (ii) the subgraph induced by the vertices assigned non-zero values has no isolated vertices. The minimum of over all such functions is called the total Roman domination number The total Roman domination stability number of a graph G with no isolated vertex, denoted by is the minimum number of vertices whose removal does not produce isolated vertices and changes the total Roman domination number of G. In this paper we present some bounds for the total Roman domination stability number of a graph, and prove that the associated decision problem is NP-hard even when restricted to bipartite graphs or planar graphs.

From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs

Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of v∈V(G). Let f:V(G)→{0,1,2} be a function and Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V1∪V2 has no isolated vertex and N(v)∩V2≠∅ for every v∈V(G)\V2. The strongly total Roman domination number of G, denoted by γtRs(G), is defined as the minimum weight ω(f)=∑x∈V(G)f(x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing γtRs(G) is NP-hard.

Algorithmic Aspects of Outer-Independent Total Roman Domination in Graphs

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Let $$G=(V,E)$$ be a graph. A function $$f : V \rightarrow \{0, 1, 2\}$$ is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex $$v \in V$$ with $$f (v) = 0$$ there is a vertex u adjacent to v with $$f (u) = 2$$ and $$\{x\in V:f(x)=0\}$$ is an independent set. The weight of f is the value $$ f(V)=\sum _{v\in V}f(v)$$ . An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every $$v\in V$$ with $$f(v)>0$$ there is a vertex u adjacent to v with $$f (u)>0$$ . The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by $$\gamma _{oiR}(G)$$ . Similarly, the outer independent total Roman domination number of G is defined, denoted by $$\gamma _{oitR}(G)$$ . In this paper, we first show that computing $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph $$G=(V,E)$$ we propose an algorithm to compute $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) in $${\mathcal {O}}(|V| )$$ time.

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.

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 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.

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.