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.

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].

The restrained k-rainbow reinforcement number of graphs

2020 ◽  
pp. 2150026
Author(s):  
Saeed Shaebani ◽  
Saeed Kosari ◽  
Leila Asgharsharghi
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
General Graphs

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple graph. A restrained [Formula: see text]-rainbow dominating function (R[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 every vertex [Formula: see text] with [Formula: see text] satisfies both of the conditions [Formula: see text] and [Formula: see text] simultaneously, where [Formula: see text] denotes the open neighborhood of [Formula: see text]. The weight of an R[Formula: see text]RDF is the value [Formula: see text]. The restrained[Formula: see text]-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an R[Formula: see text]RDF of [Formula: see text]. The restrained[Formula: see text]-rainbow reinforcement number [Formula: see text] of [Formula: see text], is defined to be the minimum number of edges that must be added to [Formula: see text] in order to decrease the restrained [Formula: see text]-rainbow domination number. In this paper, we determine the restrained [Formula: see text]-rainbow reinforcement number of some special classes of graphs. Also, we present some bounds on the restrained [Formula: see text]-rainbow reinforcement number of general graphs.

Rainbow reinforcement numbers in digraphs

2017 ◽  
Vol 10 (01) ◽  
pp. 1750004 ◽  
Author(s):  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Sharp Bounds ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Dominating Function

Let [Formula: see text] be a finite and simple digraph. A [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) of a digraph [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 set of in-neighbors of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a digraph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a [Formula: see text]RDF of [Formula: see text]. The [Formula: see text]-rainbow reinforcement number [Formula: see text] of a digraph [Formula: see text] is the minimum number of arcs that must be added to [Formula: see text] in order to decrease the [Formula: see text]-rainbow domination number. In this paper, we initiate the study of [Formula: see text]-rainbow reinforcement number in digraphs and we present some sharp bounds for [Formula: see text]. In particular, we determine the [Formula: see text]-rainbow reinforcement number of some classes of digraphs.

scholarly journals The restrained rainbow bondage number of a graph

2018 ◽  
Vol 49 (2) ◽  
pp. 115-127
Author(s):  
Jafar Amjadi ◽  
Rana Khoeilar ◽  
N. Dehgardi ◽  
Lutz Volkmann ◽  
S.M. Sheikholeslami
Keyword(s):  
Maximum Degree ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Bondage Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Dominating Function

A restrained $k$-rainbow dominating function (R$k$RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$ such that for any vertex $v \in V (G)$ with $f(v) = \emptyset$ the conditions $\bigcup_{u \in N(v)} f(u)=\{1,2,\ldots,k\}$ and $|N(v)\cap \{u\in V\mid f(u)=\emptyset\}|\ge 1$ are fulfilled, where $N(v)$ is the open neighborhood of $v$. The weight of a restrained $k$-rainbow dominating function is the value $w(f)=\sum_{v\in V}|f (v)|$. The minimum weight of a restrained $k$-rainbow dominating function of $G$ is called the restrained $k$-rainbow domination number of $G$, denoted by $\gamma_{rrk}(G)$. The restrained $k$-rainbow bondage number $b_{rrk}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $F \subseteq E(G)$ for which $\gamma_{rrk}(G-F) > \gamma_{rrk}(G)$. In this paper, we initiate the study of the restrained $k$-rainbow bondage number in graphs and we present some sharp bounds for $b_{rr2}(G)$. In addition, we determine the restrained 2-rainbow bondage number of some classes of graphs.

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.

scholarly journals k-Rainbow Domination Number of P3□Pn

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 203 ◽  
Author(s):  
Ying Wang ◽  
Xinling Wu ◽  
Nasrin Dehgardi ◽  
Jafar Amjadi ◽  
Rana Khoeilar ◽  
...  
Keyword(s):  
Positive Integer ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Dominating Function

Let k be a positive integer, and set [ k ] : = { 1 , 2 , … , k } . For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping f : V ( G ) → 2 [ k ] in such a way that, for any vertex v ∈ V ( G ) with the empty set under f, the condition ⋃ u ∈ N G ( v ) f ( u ) = [ k ] always holds, where N G ( v ) is the open neighborhood of v. The weight of kRDF f of G is the summation of values of all vertices under f. The k-rainbow domination number of G, denoted by γ r k ( G ) , is the minimum weight of a kRDF of G. In this paper, we obtain the k-rainbow domination number of grid P 3 □ P n for k ∈ { 2 , 3 , 4 } .

scholarly journals The 3-Rainbow Domination Number of the Cartesian Product of Cycles

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 65
Author(s):  
Hong Gao ◽  
Changqing Xi ◽  
Yuansheng Yang
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Cartesian Product ◽  
Power Set ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Dominating Function

We have studied the k-rainbow domination number of C n □ C m for k ≥ 4 (Gao et al. 2019), in which we present the 3-rainbow domination number of C n □ C m , which should be bounded above by the four-rainbow domination number of C n □ C m . Therefore, we give a rough bound on the 3-rainbow domination number of C n □ C m . In this paper, we focus on the 3-rainbow domination number of the Cartesian product of cycles, C n □ C m . A 3-rainbow dominating function (3RDF) f on a given graph G is a mapping from the vertex set to the power set of three colors { 1 , 2 , 3 } in such a way that every vertex that is assigned to the empty set has all three colors in its neighborhood. The weight of a 3RDF on G is the value ω ( f ) = ∑ v ∈ V ( G ) | f ( v ) | . The 3-rainbow domination number, γ r 3 ( G ) , is the minimum weight among all weights of 3RDFs on G. In this paper, we determine exact values of the 3-rainbow domination number of C 3 □ C m and C 4 □ C m and present a tighter bound on the 3-rainbow domination number of C n □ C m for n ≥ 5 .

scholarly journals Total Roman 2 -Reinforcement of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
M. Kheibari ◽  
H. Abdollahzadeh Ahangar ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Positive Weight ◽  
Minimum Number ◽  
Isolated Vertex ◽  
Dominating Function

A total Roman 2 -dominating function (TR2DF) on a graph Γ = V , E is a function l : V ⟶ 0,1,2 , satisfying the conditions that (i) for every vertex y ∈ V with l y = 0 , either y is adjacent to a vertex labeled 2 under l , or y is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF l is the value ∑ y ∈ V l y . The total Roman 2 -domination number (TR2D-number) of a graph Γ is the minimum weight of a TR2DF on Γ . The total Roman 2 -reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of 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].

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