scholarly journals The Italian Domination Numbers of Generalized Petersen Graphs P(n,3)

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 714 ◽  
Author(s):  
Hong Gao ◽  
Changqing Xi ◽  
Kun Li ◽  
Qingfang Zhang ◽  
Yuansheng Yang
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Generalized Petersen Graphs ◽  
Petersen Graphs ◽  
Dominating Function ◽  
Domination Numbers

An Italian dominating function of G is a function f : V ( G ) → { 0 , 1 , 2 } , for every vertex v such that f ( v ) = 0 , it holds that ∑ u ∈ N ( v ) f ( u ) ≥ 2 . The Italian domination number γ I ( G ) is the minimum weight of an Italian dominating function on G. In this paper, we determine the exact values of the Italian domination numbers of P ( n , 3 ) .

scholarly journals The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

2018 ◽  
Author(s):  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao ◽  
Jia-Bao Liu
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

A double Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2, 3} 2 with the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for 3 which f(v) = 3 or two vertices v1 and v2 for which f(v1) = f(v2) = 2, and every vertex u for which 4 f(u) = 1 is adjacent to at least one vertex v for which f(v) ≥ 2. The weight of a double Roman dominating function f is the value w(f) = ∑u∈V(G) 5 f(u). The minimum weight over all double 6 Roman dominating functions on a graph G is called the double Roman domination number γdR(G) 7 of G. In this paper we determine the exact value of the double Roman domination number of the 8 generalized Petersen graphs P(n, 2) by using a discharging approach.

scholarly journals The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 206 ◽  
Author(s):  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao ◽  
Yongsheng Rao ◽  
Jia-Bao Liu
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to { 0 , 1 , 2 , 3 } in such a way that a vertex u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a DRDF on a graph G is called the double Roman domination number γ d R ( G ) of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs P ( n , 2 ) by using a discharging approach.

scholarly journals Double Roman Graphs in P(3k, k)

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 336
Author(s):  
Zehui Shao ◽  
Rija Erveš ◽  
Huiqin Jiang ◽  
Aljoša Peperko ◽  
Pu Wu ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Sharp Lower Bound ◽  
Dominating Function

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

scholarly journals On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 119
Author(s):  
Darja Rupnik Poklukar ◽  
Janez Žerovnik
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Dominating Function

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

Twin signed total Roman domination numbers in digraphs

2018 ◽  
Vol 11 (03) ◽  
pp. 1850034 ◽  
Author(s):  
J. Amjadi ◽  
M. Soroudi
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

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.

scholarly journals On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1860
Author(s):  
Rija Erveš ◽  
Janez Žerovnik
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs ◽  
Special Case ◽  
Domination Numbers

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.

scholarly journals Lower bounds on the Roman and independent Roman domination numbers

2016 ◽  
Vol 10 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Mustapha Chellali ◽  
Teresa Haynes ◽  
Stephen Hedetniemi
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Independent Domination ◽  
Dominating Function ◽  
Domination Numbers

A Roman dominating function (RDF) on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the sum f(V) = ?v?V f(v), and the minimum weight of a Roman dominating function f is the Roman domination number ?R(G). An RDF f is called an independent Roman dominating function (IRDF) if the set of vertices assigned positive values under f is independent. The independent Roman domination number iR(G) is the minimum weight of an IRDF on G. We show that for every nontrivial connected graph G with maximum degree ?, ?R(G)? ?+1/??(G) and iR(G) ? i(G) + ?(G)/?, where ?(G) and i(G) are, respectively, the domination and independent domination numbers of G. Moreover, we characterize the connected graphs attaining each lower bound. We give an additional lower bound for ?R(G) and compare our two new bounds on ?R(G) with some known lower bounds.

Rainbow edge domination numbers in graphs

2019 ◽  
Vol 12 (07) ◽  
pp. 2050004
Author(s):  
H. Abdollahzadeh Ahangar ◽  
H. Jahani ◽  
N. Jafari Rad
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Sharp Bounds ◽  
Domination In Graphs ◽  
Dominating Function ◽  
Domination Numbers

A 2-rainbow edge dominating function (2REDF) of a graph [Formula: see text] is a function [Formula: see text] from the edge set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any edge [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 2REDF [Formula: see text] is the value [Formula: see text]. The minimum weight of a 2REDF is the 2-rainbow edge domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of 2-rainbow edge domination in graphs. We present various sharp bounds, exact values and characterizations for the 2-rainbow edge domination number of a graph.

scholarly journals Twin signed Roman domination numbers in directed graphs

2016 ◽  
Vol 47 (3) ◽  
pp. 357-371 ◽  
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Asghar Bodaghli ◽  
Lutz Volkmann
Keyword(s):  
Directed Graphs ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin signed Roman domination number $\gamma_{sR}^*(D)$ of $D$ is the minimum weight of an TSRDF on $D$. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on $\gamma_{sR}^*(D)$. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

On Double Domination Numbers of Generalized Petersen Graphs

2016 ◽  
Vol 13 (10) ◽  
pp. 6514-6518
Author(s):  
Minhong Sun ◽  
Zehui Shao
Keyword(s):  
Programming Model ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Size ◽  
Generalized Petersen Graphs ◽  
Small Parameters ◽  
Petersen Graphs ◽  
Double Domination ◽  
Domination Numbers

A (total) double dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is (total) dominated by at least 2 vertices in S. The (total) double domination number of G is the minimum size of a (total) double dominating set of G. We determine the total double domination numbers and give upper bounds for double domination numbers of generalized Petersen graphs. By applying an integer programming model for double domination numbers of a graph, we have determined some exact values of double domination numbers of these generalized Petersen graphs with small parameters. The result shows that the given upper bounds match these exact values.

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