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.

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 the Unique Response Roman Domination Numbers of Graphs

2016 ◽  
Vol 13 (10) ◽  
pp. 7362-7365
Author(s):  
Yong Li ◽  
Qiong Li ◽  
Jian He ◽  
Xinruan Fan ◽  
Zhaoheng Ding
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

Let G be a graph with vertex set V(G). A function f: V(G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V(G), where Vi = {V∈V(G) | f(V) = i} for i = 0, 1, 2, is a Roman dominating function if x ∈ V0 implies |N(x)∩V2|≥ 1. It is a unique response Roman function if x ∈ V0 implies |N(x) ≥ V2|≤ 1 and x ∈ V1 ∪ V2 implies that |N(x) ∩ V2| = 0. A function f: V(G) → {0, 1, 2} is a unique response Roman dominating function if it is both a unique response Roman function and a Roman dominating function. The unique response Roman domination number, denoted by uR(G), of G is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination of graphs, and provide some graphs whose unique response Roman domination number equals to the independent Roman domination number.

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.

Trees with independent Roman domination number twice the independent domination number

2015 ◽  
Vol 07 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Mustapha Chellali ◽  
Nader Jafari Rad
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Independent Domination ◽  
Open Question ◽  
Dominating Function

A Roman dominating function (RDF) 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]. The weight of a RDF [Formula: see text] is the value [Formula: see text]. The Roman domination number, [Formula: see text], of [Formula: see text] is the minimum weight of a RDF on [Formula: see text]. An RDF [Formula: see text] is called an independent Roman dominating function (IRDF) if the set [Formula: see text] is an independent set. The independent Roman domination number, [Formula: see text], is the minimum weight of an IRDF on [Formula: see text]. In this paper, we study trees with independent Roman domination number twice their independent domination number, answering an open question.

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 The Roman bondage number of a digraph

2016 ◽  
Vol 47 (4) ◽  
pp. 421-431
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Nasrin Dehgardi ◽  
Lutz Volkmann ◽  
Dirk Meierling
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

Let $D=(V,A)$ be a finite and simple digraph. A  Roman dominating function on $D$ is a labeling $f:V (D)\rightarrow \{0, 1, 2\}$ such that every vertex with label 0 has an in-neighbor with label 2. The weight of an RDF $f$ is the value $\omega(f)=\sum_{v\in V}f (v)$. The minimum weight of a Roman dominating function on a digraph $D$ is called the Roman domination number, denoted by $\gamma_{R}(D)$. The Roman bondage number $b_{R}(D)$ of a digraph $D$ with maximum out-degree at least two is the minimum cardinality of all sets $A'\subseteq A$ for which $\gamma_{R}(D-A')>\gamma_R(D)$. In this paper, we initiate the study of the Roman bondage number of a digraph. We determine the Roman bondage number in several classes of digraphs and give some sharp bounds.

Bounds on the co-Roman domination number in graphs

2019 ◽  
Vol 13 (08) ◽  
pp. 2050140
Author(s):  
N. Dehgardi ◽  
S. M. Sheikholeslami ◽  
M. Soroudi ◽  
L. Volkmann
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Positive Weight ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

Let [Formula: see text] be a graph and let [Formula: see text] be a function. A vertex [Formula: see text] is protected with respect to [Formula: see text], if [Formula: see text] or [Formula: see text] and [Formula: see text] is adjacent to a vertex of positive weight. The function [Formula: see text] is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex in [Formula: see text] is protected, and (ii) each [Formula: see text] with positive weight has a neighbor [Formula: see text] with [Formula: see text] such that the function [Formula: see text], defined by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], has no unprotected vertex. The weight of [Formula: see text] is [Formula: see text]. The co-Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a co-Roman dominating function on [Formula: see text]. In this paper, we present some new sharp bounds on [Formula: see text]. Some of our results improve the previous bounds.

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.

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.

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