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.

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.

scholarly journals Quadruple Roman Domination in Trees

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1318
Author(s):  
Zheng Kou ◽  
Saeed Kosari ◽  
Guoliang Hao ◽  
Jafar Amjadi ◽  
Nesa Khalili
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Upper And Lower Bounds ◽  
Discrete Mathematics ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

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.

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.

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.

The signed total Roman domatic number of a digraph

2018 ◽  
Vol 10 (02) ◽  
pp. 1850020 ◽  
Author(s):  
J. Amjadi
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Domatic Number ◽  
Vertex Set ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Dominating Function

Let [Formula: see text] be a finite simple digraph with vertex set [Formula: see text]. A signed total Roman dominating function (STRDF) on a digraph [Formula: see text] is a function [Formula: see text] such that (i) [Formula: see text] for every [Formula: see text], where [Formula: see text] consists of all inner neighbors of [Formula: see text], and (ii) every vertex [Formula: see text] for which [Formula: see text] has an inner neighbor [Formula: see text] for which [Formula: see text]. The weight of an STRDF [Formula: see text] is [Formula: see text]. The signed total Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of an STRDF on [Formula: see text]. A set [Formula: see text] of distinct STRDFs on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a signed total Roman dominating family (STRD family) (of functions) on [Formula: see text]. The maximum number of functions in an STRD family on [Formula: see text] is the signed total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we initiate the study of signed total Roman domatic number in digraphs and we present some sharp bounds for [Formula: see text]. In addition, we determine the signed total Roman domatic number of some classes of digraphs.

Independent Roman domination and 2-independence in trees

2018 ◽  
Vol 10 (04) ◽  
pp. 1850052
Author(s):  
J. Amjadi ◽  
S. M. Sheikholeslami ◽  
M. Valinavaz ◽  
N. Dehgardi
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Two Parameters ◽  
Dominating Function

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. A Roman dominating function [Formula: see text] is called an independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function [Formula: see text] is the value [Formula: see text]. The independent Roman domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an independent Roman dominating function on [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text]. The maximum cardinality of a 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. These two parameters are incomparable in general, however, we show that for any tree [Formula: see text], [Formula: see text] and we characterize all trees attaining the equality.

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.

scholarly journals Vertex-Edge Roman Domination

2021 ◽  
Vol 45 (5) ◽  
pp. 685-698
Author(s):  
H. NARESH KUMAR ◽  
◽  
Y. B. VENKATAKRISHNAN
Keyword(s):  
Lower Bound ◽  
Minimum Weight ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Np Complete ◽  
Dominating Function

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number γveR(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γveR(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γveR(T) = 2γ′(T), where γ′(T) is the edge domination number of T.

scholarly journals Roman Domination of Some Chemical Graphs

2021 ◽  
pp. 556-562
Author(s):  
Pallavi Sangolli ◽  
Manjula C. Gudgeri ◽  
. Varsha ◽  
Shailaja S. Shirkol
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Wiener Index ◽  
Roman Domination ◽  
Zagreb Index ◽  
Dna Structures ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Domination In Graphs ◽  
Dominating Function

The concept of Domination in graphs has application to the study of DNA structures. For investigating the chemical and physical properties, several topological indices used are Wiener index, Randic index, Zagreb index, Kier & Hall index that depends on vertex degree and distance sum, and have been used extensively for QSAR and QSPR studies. A Roman Dominating Function of G is function f: V→ {0, 1, 2} such that every vertex v for which f (v) = 0 has a neighbor u with f(u) = 2. The weight of a Roman dominating function f is w (f) =   . The Roman domination number of a graph G is denoted by (G) and is the minimum weight of all possible Roman dominating functions. In this paper, we find Roman domination number of some chemicals graphs such as saturated hydrocarbons and unsaturated hydrocarbons, hexagonal chain, pyrene, Hexabenzocoronene, H-Phenylenic nanotube and N-Napthelenic nanotube.

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