scholarly journals Total Roman {3}-domination in Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 268
Author(s):  
Zehui Shao ◽  
Doost Ali Mojdeh ◽  
Lutz Volkmann
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Minimum Weight ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Additional Property ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Dominating Function

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.

scholarly journals Some progress on the mixed Roman domination in graphs

2020 ◽  
Author(s):  
Hossein Abdollahzadeh Ahangar ◽  
Jafar Amjadi ◽  
Mustapha Chellali ◽  
S. Kosari ◽  
Vladimir Samodivkin ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Domination In Graphs ◽  
Dominating Function

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A mixed Roman dominating function (MRDF) of $G$ is a function $f:V\cup E\rightarrow \{0,1,2\}$ satisfying the condition that every element $x\in V\cup E$ for which $f(x)=0$ is adjacent or incident to at least one element $% y\in V\cup E$ for which $f(y)=2$. The weight of a mixed Roman dominating function $f$ is $\omega (f)=\sum_{x\in V\cup E}f(x)$. The mixed Roman domination number $\gamma _{R}^{\ast }(G)$ of $G$ is the minimum weight of a mixed Roman dominating function of $G$. We first show that the problem of computing $\gamma _{R}^{\ast }(G)$ is NP-complete for bipartite graphs and then we present upper and lower bounds on the mixed Roman domination number, some of them are for the class of trees.

scholarly journals New bounds on the rainbow domination subdivision number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 615-622 ◽  
Author(s):  
Mohyedin Falahat ◽  
Seyed Sheikholeslami ◽  
Lutz Volkmann
Keyword(s):  
Connected Graph ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Domination Subdivision Number ◽  
Dominating Function

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v ? V(G) with f (v) = ? the condition Uu?N(v) f(u)= {1,2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ?(f) = ?v?V |f(v)|. The 2-rainbow domination number of a graph G, denoted by r2(G), is the minimum weight of a 2RDF of G. The 2-rainbow domination subdivision number sd?r2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-rainbow domination number. In this paper we prove that for every simple connected graph G of order n ? 3, sd?r2(G)? 3 + min{d2(v)|v?V and d(v)?2} where d2(v) is the number of vertices of G at distance 2 from v.

On the rainbow domination subdivision numbers in graphs

2016 ◽  
Vol 09 (01) ◽  
pp. 1650018 ◽  
Author(s):  
N. Dehgardi ◽  
M. Falahat ◽  
S. M. Sheikholeslami ◽  
Abdollah Khodkar
Keyword(s):  
Connected Graph ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Domination Subdivision Number ◽  
Dominating Function

A [Formula: see text]-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of G. The [Formula: see text]-rainbow domination subdivision number [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the 2-rainbow domination number. It is conjectured that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. In this paper, we first prove this conjecture for some classes of graphs and then we prove that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text].

Isolate Roman domination in graphs

2021 ◽  
pp. 2150131
Author(s):  
Davood Bakhshesh
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Chordal Graphs ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Domination In Graphs ◽  
Dominating Function

Let [Formula: see text] be a graph with the vertex set [Formula: see text]. A function [Formula: see text] is called a Roman dominating function of [Formula: see text], if every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text]. The weight of a Roman dominating function [Formula: see text] is equal to [Formula: see text]. The minimum weight of a Roman dominating function of [Formula: see text] is called the Roman domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of a variant of Roman dominating functions. A function [Formula: see text] is called an isolate Roman dominating function of [Formula: see text], if [Formula: see text] is a Roman dominating function and there is a vertex [Formula: see text] with [Formula: see text] which is not adjacent to any vertex [Formula: see text] with [Formula: see text]. The minimum weight of an isolate Roman dominating function of [Formula: see text] is called the isolate Roman domination number of [Formula: see text], denoted by [Formula: see text]. We present some upper bound on the isolate Roman domination number of a graph [Formula: see text] in terms of its Roman domination number and its domination number. Moreover, we present some classes of graphs [Formula: see text] with [Formula: see text]. Finally, we show that the decision problem associated with the isolate Roman dominating functions is NP-complete for bipartite graphs and chordal graphs.

Independent 2-rainbow domination in trees

2015 ◽  
Vol 08 (02) ◽  
pp. 1550035 ◽  
Author(s):  
J. Amjadi ◽  
N. Dehgardi ◽  
N. Mohammadi ◽  
S. M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Sharp Bound ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Dominating Function ◽  
Domination Numbers

A 2-rainbow dominating function (2RDF) on a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃u∈N(v)f(u) = {1, 2} is fulfilled. A 2RDF f is independent 2-rainbow dominating function (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value ω(f) = ∑v∈V |f(v)|. The 2-rainbow domination number γr2(G) (respectively, the independent 2-rainbow domination number ir2(G)) is the minimum weight of a 2RDF (respectively, I2RDF) on G. M. Chellali and N. Jafari Rad [Independent 2-rainbow domination in graphs, to appear in J. Combin. Math. Combin. Comput.] have studied the independent 2-rainbow domination numbers in graphs and posed the following problem: Find a sharp bound for ir2(T) in terms of the order of a tree T. In this paper we prove that for every tree T of order n ≥ 3, [Formula: see text].

Bounds on the signed total Roman 2-domination in graphs

2020 ◽  
Vol 12 (01) ◽  
pp. 2050013 ◽  
Author(s):  
R. Khoeilar ◽  
L. Shahbazi ◽  
S. M. Sheikholeslami ◽  
Zehui Shao
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Finite Graph ◽  
Sharp Bounds ◽  
Vertex Set ◽  
Number Formula ◽  
Domination In Graphs ◽  
Dominating Function

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.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

Resolving Roman domination in graphs

2021 ◽  
Author(s):  
P. Roushini Leely Pushpam ◽  
B. Mahavir ◽  
M. Kamalam
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

Let [Formula: see text] be a graph and [Formula: see text] be a Roman dominating function defined on [Formula: see text]. Let [Formula: see text] be some ordering of the vertices of [Formula: see text]. For any [Formula: see text], [Formula: see text] is defined by [Formula: see text]. If for all [Formula: see text], [Formula: see text], we have [Formula: see text], that is [Formula: see text], for some [Formula: see text], then [Formula: see text] is called a resolving Roman dominating function (RDF) on [Formula: see text]. The weight of a resolving RDF [Formula: see text] on [Formula: see text] is [Formula: see text]. The minimum weight of a resolving RDF on [Formula: see text] is called the resolving Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. A resolving RDF on [Formula: see text] with weight [Formula: see text] is called a [Formula: see text]-function on [Formula: see text]. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order.

Total k-rainbow reinforcement number in graphs

2020 ◽  
pp. 2050101
Author(s):  
L. Shahbazi ◽  
H. Abdollahzadeh Ahangar ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Complete Bipartite Graphs ◽  
Dominating Function

Let [Formula: see text] be an integer, and let [Formula: see text] be a graph. A k-rainbow dominating function (or [Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text] such that for very [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. A k-rainbow dominating function [Formula: see text] in a graph with no isolated vertex is called a total k-rainbow dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertices. The total k-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of the total [Formula: see text]-rainbow dominating function on [Formula: see text]. The total k-rainbow reinforcement number of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges that must be added to [Formula: see text] in order to decrease the total k-rainbow domination number. In this paper, we investigate the properties of total [Formula: see text]-rainbow reinforcement number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total [Formula: see text]-rainbow reinforcement number of some classes of graphs including paths, cycles and complete bipartite graphs.

Critical concept for double Roman domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050020
Author(s):  
S. Nazari-Moghaddam ◽  
L. Volkmann
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Critical Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

A double Roman dominating function (DRDF) 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 two vertices assigned a [Formula: see text] or to at least one vertex assigned a [Formula: see text] and (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number [Formula: see text] equals the minimum weight of a DRDF on [Formula: see text] The concept of criticality with respect to various operations on graphs has been studied for several domination parameters. In this paper, we study the concept of criticality for double Roman domination in graphs. In addition, we characterize double Roman domination edge super critical graphs and we will give several characterizations for double Roman domination vertex (edge) critical graphs.

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