Signed domination and Mycielski’s structure in graphs

Let G = (V, E) be a graph. The function f : V(G) → {−1, 1} is a signed dominating function if for every vertex v ∈ V(G), ∑x∈NG[v] f(x)≥1. The value of ω(f) = ∑x∈V(G) f(x) is called the weight of f. The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete bipartite graph.

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Signed Domination Number of the Directed Cylinder

Symmetry ◽

10.3390/sym11121443 ◽

2019 ◽

Vol 11 (12) ◽

pp. 1443

Author(s):

Haichao Wang ◽

Hye Kyung Kim

Keyword(s):

Minimum Weight ◽

Domination Number ◽

Cartesian Product ◽

Directed Cycle ◽

Directed Path ◽

Signed Domination ◽

Signed Domination Number ◽

Positive Integers ◽

Dominating Function

In a digraph D = ( V ( D ) , A ( D ) ) , a two-valued function f : V ( D ) → { - 1 , 1 } defined on the vertices of D is called a signed dominating function if f ( N - [ v ] ) ≥ 1 for every v in D. The weight of a signed dominating function is f ( V ( D ) ) = ∑ v ∈ V ( D ) f ( v ) . The signed domination number γ s ( D ) is the minimum weight among all signed dominating functions of D. Let P m × C n be the Cartesian product of directed path P m and directed cycle C n . In this paper, the exact value of γ s ( P m × C n ) is determined for any positive integers m and n.

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Twin signed total Roman domination numbers in digraphs

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500341 ◽

2018 ◽

Vol 11 (03) ◽

pp. 1850034 ◽

Cited By ~ 1

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.

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Signed domination number of hypercubes

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092250029x ◽

2021 ◽

Author(s):

B. ShekinahHenry ◽

Y. S. Irine Sheela

Keyword(s):

Domination Number ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Signed Domination ◽

Vertex Set ◽

Number Formula ◽

Signed Domination Number

The [Formula: see text]-cube graph or hypercube [Formula: see text] is the graph whose vertex set is the set of all [Formula: see text]-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one co-ordinate. The purpose of the paper is to investigate the signed domination number of this hypercube graphs. In this paper, signed domination number [Formula: see text]-cube graph for odd [Formula: see text] is found and the lower and upper bounds of hypercube for even [Formula: see text] are found.

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Outer independent rainbow dominating functions in graphs

Opuscula Mathematica ◽

10.7494/opmath.2020.40.5.599 ◽

2020 ◽

Vol 40 (5) ◽

pp. 599-615

Author(s):

Zhila Mansouri ◽

Doost Ali Mojdeh

Keyword(s):

Minimum Weight ◽

Vertex Cover ◽

Domination Number ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Rainbow Domination ◽

Dominating Function

A 2-rainbow dominating function (2-rD function) of a graph $G=(V,E)$ is a function $f:V(G)\rightarrow\{\emptyset,\{1\},\{2\},\{1,2\}\}$ having the property that if $f(x)=\emptyset$, then $f(N(x))=\{1,2\}$. The 2-rainbow domination number $\gamma_{r2}(G)$ is the minimum weight of $\sum_{v\in V(G)}|f(v)|$ taken over all 2-rainbow dominating functions $f$. An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph $G$ is a 2-rD function $f$ for which the set of all $v\in V(G)$ with $f(v)=\emptyset$ is independent. The outer independent 2-rainbow domination number $\gamma_{oir2}(G)$ is the minimum weight of an OI2-rD function of $G$. In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on $\gamma_{oir2}(G)$. Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair $(a,b)$ is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if $a+1\leq b\leq 2a$.

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Lower bounds on the Roman and independent Roman domination numbers

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm151112023c ◽

2016 ◽

Vol 10 (1) ◽

pp. 65-72 ◽

Cited By ~ 7

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.

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Rainbow edge domination numbers in graphs

Asian-European Journal of Mathematics ◽

10.1142/s1793557120500047 ◽

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.

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

10.20944/preprints201807.0381.v1 ◽

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.

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Twin signed Roman domination numbers in directed graphs

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.47.2016.2035 ◽

2016 ◽

Vol 47 (3) ◽

pp. 357-371 ◽

Cited By ~ 1

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.

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Twin signed k-domination numbers in directed graphs

Filomat ◽

10.2298/fil1720367d ◽

2017 ◽

Vol 31 (20) ◽

pp. 6367-6378

Author(s):

Nasrin Dehgardi ◽

Maryam Atapour ◽

Abdollah Khodkar

Keyword(s):

Directed Graph ◽

Directed Graphs ◽

Domination Number ◽

Signed Domination ◽

Dominating Function ◽

Domination Numbers

Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.

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Bounds for signed double Roman k-domination in trees

RAIRO - Operations Research ◽

10.1051/ro/2018043 ◽

2019 ◽

Vol 53 (2) ◽

pp. 627-643 ◽

Cited By ~ 2

Author(s):

Hong Yang ◽

Pu Wu ◽

Sakineh Nazari-Moghaddam ◽

Seyed Mahmoud Sheikholeslami ◽

Xiaosong Zhang ◽

...

Keyword(s):

Minimum Weight ◽

Domination Number ◽

Finite Graph ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Vertex Set ◽

Dominating Function

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this paper, we investigate the signed double Roman k-domination number of trees. In particular, we present lower and upper bounds on γksdR(T) for 2 ≤ k ≤ 6 and classify all extremal trees.

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