BOUNDS ON THE DOMINATION NUMBER OF PERMUTATION GRAPHS

2009 ◽  
Vol 10 (03) ◽  
pp. 205-217 ◽  
Author(s):  
WEIZHEN GU ◽  
KIRSTI WASH
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Permutation Graph ◽  
Original Graph ◽  
Permutation Graphs ◽  
Positive Integers

For a graph G with n vertices and a permutation α on V(G), a permutation graph Pα(G) is obtained from two identical copies of G by adding an edge between v and α(V) for any v ϵ V(G). Let γ(G) be the domination number of a graph G. It has been shown that γ(G) ≤ γ(Pα(G) ≤ 2γ(G) for any permutation α on V(G). In this paper, we investigate specific graphs for which there exists a permutation α such that γ(Pα(G)) ≻ γ(G) in terms of the domination number of G or the maximum degree of G. Additionally, we construct a class of graphs for which the domination number of any permutation graph is twice the domination number of the original graph, as well as explore finding a specific graph G and permutation α for any two positive integers a and b with a ≤ b ≤ 2a, to have γ(G) = a and γ(Pα(G)) = b.

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

The intersection graph of ideals of ℤm

2019 ◽  
Vol 11 (04) ◽  
pp. 1950037 ◽  
Author(s):  
S. Khojasteh
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Independence Number ◽  
Domination Number ◽  
Intersection Graph ◽  
Graph Structure ◽  
Chromatic Index ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Positive Integers

Let [Formula: see text] be an integer, and let [Formula: see text] be the set of all non-zero proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] be a [Formula: see text]-module. In this paper, we study a kind of graph structure of [Formula: see text], denoted by [Formula: see text]. It is the undirected graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Clearly, [Formula: see text]. Let [Formula: see text] and [Formula: see text], where [Formula: see text]’s are distinct primes, [Formula: see text]’s are positive integers, [Formula: see text]’s are non-negative integers, and [Formula: see text] for [Formula: see text] and let [Formula: see text], [Formula: see text]. The cardinality of [Formula: see text] is denoted by [Formula: see text]. Also, let [Formula: see text], [Formula: see text] and [Formula: see text] denote the independence number, the domination number and the set of all isolated vertices of [Formula: see text], respectively. We prove that [Formula: see text] and we show that if [Formula: see text] is not a null graph, then [Formula: see text] and [Formula: see text] We also compute some of its numerical invariants, namely maximum degree and chromatic index. Among other results, we determine all integer numbers [Formula: see text] and [Formula: see text] for which [Formula: see text] is Eulerian.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

scholarly journals On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

A note on the 2-rainbow bondage numbers in graphs

2016 ◽  
Vol 09 (01) ◽  
pp. 1650013
Author(s):  
L. Asgharsharghi ◽  
S. M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Planar Graph ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Number Formula ◽  
Appl Math

A 2-rainbow dominating function (2RDF) of a graph [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. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of [Formula: see text]. The rainbow bondage number [Formula: see text] of a graph [Formula: see text] with maximum degree at least two is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. Dehgardi, Sheikholeslami and Volkmann, [The [Formula: see text]-rainbow bondage number of a graph, Discrete Appl. Math. 174 (2014) 133–139] proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper, we generalize their result for graphs which admit a [Formula: see text]-cell embedding on a surface with non-negative Euler characteristic.

scholarly journals ON THE ROMAN BONDAGE NUMBER OF A GRAPH

2013 ◽  
Vol 05 (01) ◽  
pp. 1350001 ◽  
Author(s):  
A. BAHREMANDPOUR ◽  
FU-TAO HU ◽  
S. M. SHEIKHOLESLAMI ◽  
JUN-MING XU
Keyword(s):  
Decision Problem ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Np Hard ◽  
Roman Domination ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Roman Domination Number ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a RDF is the value f(V(G)) = Σu∈V(G) f(u). The minimum weight of a RDF on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G - E′) > γR(G). In this paper, we first show that the decision problem for determining bR(G) is NP-hard even for bipartite graphs and then we establish some sharp bounds for bR(G) and characterizes all graphs attaining some of these bounds.

scholarly journals On the upper geodetic global domination number of a graph

2020 ◽  
Vol 39 (6) ◽  
pp. 1627-1647
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
General Properties ◽  
Geodetic Set ◽  
Positive Integers ◽  
Global Domination Number

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

scholarly journals Signed strong Roman domination in graphs

2017 ◽  
Vol 48 (2) ◽  
pp. 135-147 ◽  
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar ◽  
Leila Asgharsharghi
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Simple Graph ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Closed Neighborhood ◽  
Domination In Graphs ◽  
Dominating Function

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximum degree $\Delta$. A signed strong Roman dominating function (abbreviated SStRDF) on a graph $G$ is a function $f:V\to \{-1,1,2,\ldots,\lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the conditions that (i) for every vertex $v$ of $G$, $\sum_{u\in N[v]} f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$ and (ii) every vertex $v$ for which $f(v)=-1$ is adjacent to at least one vertex $u$ for which $f(u)\ge 1+\lceil\frac{1}{2}|N(u)\cap V_{-1}|\rceil$, where $V_{-1}=\{v\in V \mid f(v)=-1\}$. The minimum of the values $\sum_{v\in V} f(v)$, taken over all signed strong Roman dominating functions $f$ of $G$, is called the signed strong Roman domination number of $G$ and is denoted by $\gamma_{ssR}(G)$. In this paper we initiate the study of the signed strong Roman domination in graphs and present some (sharp) bounds for this parameter.

scholarly journals Line Graphs of Monogenic Semigroup Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu ◽  
Sedat Pak
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Minimum Degree ◽  
Line Graph ◽  
Domination Number ◽  
Clique Number ◽  
Line Graphs ◽  
Monogenic Semigroup ◽  
Graph Properties ◽  
Graph Parameters

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

scholarly journals Further Analysis on the Total Number of Subtrees of Trees

10.37236/2186 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Shuchao Li ◽  
Shujing Wang
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Binary Trees

When considering the total number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to some other graphical indices in applications. Along this line, it is interesting to study that over some types of trees with a given order, which trees minimize or maximize this number.  Here are our main results:  (1) The extremal tree which minimizes the total number of subtrees among $n$-vertex trees with $k$ pendants is characterized.  (2) The extremal tree which maximizes (resp. minimizes) the total number of subtrees among $n$-vertex trees with a given bipartition is characterized.  (3) The extremal tree which minimizes the total number of subtrees among the set of all $q$-ary trees with $n$ non-leaf vertices is identified.  (4) The extremal $n$-vertex tree with given domination number maximizing the total number of subtrees is characterized.

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