scholarly journals V L Reciprocal Status Index and Co-Index of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Veerebradiah Lokesha ◽  
Suvarna ◽  
Ahmet Sinan Cevik ◽  
Ismail Naci Cangul
Keyword(s):  
Sharp Bounds ◽  
Vertex Set ◽  
Status Index

For the vertex set V G of a graph G , the sum of reciprocals of the breadth (distances) between the vertex v ∈ V G and whole other remaining vertices of G is called reciprocal status of v . In this study, first of all, we introduced the V L reciprocal status index and V L reciprocal status co-index of a graph G . Later, we exposed some sharp bounds for these indices. Furthermore, we determined the V L reciprocal status (co-)index over some standard graphs. Finally, we presented correlations between V L reciprocal status index and some properties of Butane derivatives via a table and illustrated with a figure.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

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.

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 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.

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.

Resolving domination number of graphs

2019 ◽  
Vol 11 (06) ◽  
pp. 1950071
Author(s):  
Ridho Alfarisi ◽  
Dafik ◽  
Arika Indah Kristiana
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Vertex Set

For a set [Formula: see text] of vertices of a graph [Formula: see text], the representation multiset of a vertex [Formula: see text] of [Formula: see text] with respect to [Formula: see text] is [Formula: see text], where [Formula: see text] is a distance between of the vertex [Formula: see text] and the vertices in [Formula: see text] together with their multiplicities. The set [Formula: see text] is a resolving set of [Formula: see text] if [Formula: see text] for every pair [Formula: see text] of distinct vertices of [Formula: see text]. The minimum resolving set [Formula: see text] is a multiset basis of [Formula: see text]. If [Formula: see text] has a multiset basis, then its cardinality is called multiset dimension, denoted by [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is a dominating set for [Formula: see text] if every vertex of [Formula: see text] that is not in [Formula: see text] is adjacent to some vertex of [Formula: see text]. The minimum cardinality of the dominating set is a domination number, denoted by [Formula: see text]. A vertex set of some vertices in [Formula: see text] that is both resolving and dominating set is a resolving dominating set. The minimum cardinality of resolving dominating set is called resolving domination number, denoted by [Formula: see text]. In our paper, we investigate and establish sharp bounds of the resolving domination number of [Formula: see text] and determine the exact value of some family graphs.

scholarly journals Twin signed Roman domatic numbers in digraphs

2017 ◽  
Vol 48 (3) ◽  
pp. 265-272
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Lutz Volkmann
Keyword(s):  
Sharp Bounds ◽  
Domatic Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Dominating Function

Let $D$ be a finite simple digraph with vertex set $V(D)$. A twin signed Roman dominating function 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 $f(v)=f(w)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct twin signed Roman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each $v\in V(D)$, is called a twin signed Roman dominating family (of functions) on $D$. The maximum number of functions in a twin signed Roman dominating family on $D$ is the twin signed Roman domatic number of $D$, denoted by $d_{sR}^*(D)$. In this paper, we initiate the study of the twin signed Roman domatic number in digraphs and we present some sharp bounds on $d_{sR}^*(D)$. In addition, we determine the twin signed Roman domatic number of some classes of digraphs.

Bounds on the signed total Roman 2-domination in graphs

2020 ◽  
Vol 12 (01) ◽  
pp. 2050013 ◽  
Author(s):  
R. Khoeilar ◽  
L. Shahbazi ◽  
S. M. Sheikholeslami ◽  
Zehui Shao
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Finite Graph ◽  
Sharp Bounds ◽  
Vertex Set ◽  
Number Formula ◽  
Domination In Graphs ◽  
Dominating Function

Let [Formula: see text] be an integer and [Formula: see text] be a simple and finite graph with vertex set [Formula: see text]. A signed total Roman [Formula: see text]-dominating function (STR[Formula: see text]DF) 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 one vertex [Formula: see text] with [Formula: see text] and (ii) [Formula: see text] holds for any vertex [Formula: see text]. The weight of an STR[Formula: see text]DF [Formula: see text] is [Formula: see text] and the minimum weight of an STR[Formula: see text]DF is the signed total Roman [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, we establish some sharp bounds on the signed total Roman 2-domination number.

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.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

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