On Dominating Sets and Independent Sets of Graphs

1999 ◽  
Vol 8 (6) ◽  
pp. 547-553 ◽  
Author(s):  
JOCHEN HARANT ◽  
ANJA PRUCHNEWSKI ◽  
MARGIT VOIGT
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Dominating Sets ◽  
Integral Vector ◽  
Vertex Set ◽  
Usual Concept ◽  
Dimensional Cube

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 [les ] ki [les ] di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ V[setmn ]Dk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) [mid ] pi ∈ ℝ, 0 [les ] pi [les ] 1, i = 1, …, n}. An [Oscr ](Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with [mid ]Dk[mid ][les ]fk(p).

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.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals A New Upper Bound on the Total Domination Number of a Graph

10.37236/983 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal L}$; it is a path. If $|P| \equiv 0 \, ( {\rm mod} \, 4)$ and either the two ends of $P$ are adjacent in $G$ to the same large vertex or the two ends of $P$ are adjacent to different, but adjacent, large vertices in $G$, we call $P$ a $0$-path. If $|P| \ge 5$ and $|P| \equiv 1 \, ( {\rm mod} \, 4)$ with the two ends of $P$ adjacent in $G$ to the same large vertex, we call $P$ a $1$-path. If $|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call $P$ a $3$-path. For $i \in \{0,1,3\}$, we denote the number of $i$-paths in $G$ by $p_i$. We show that the total domination number of $G$ is at most $(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if $G$ is a graph of order $n$ with minimum degree at least three, then the total domination of $G$ is at most $n/2$. It also generalizes a result by Lam and Wei stating that if $G$ is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-$2$ vertices, then the total domination of $G$ is at most $n/2$.

Some results for the two disjoint connected dominating sets problem

2019 ◽  
Vol 11 (06) ◽  
pp. 1950065
Author(s):  
Xianliang Liu ◽  
Zishen Yang ◽  
Wei Wang
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Sufficient Condition ◽  
Dominating Sets ◽  
Cubic Graph ◽  
Connected Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Dominating Set Problem

As a variant of minimum connected dominating set problem, two disjoint connected dominating sets (DCDS) problem is to ask whether there are two DCDS [Formula: see text] in a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], and if not, how to add an edge subset with minimum cardinality such that the new graph has a pair of DCDS. The two DCDS problem is so hard that it is NP-hard on trees. In this paper, if the vertex set [Formula: see text] of a connected graph [Formula: see text] can be partitioned into two DCDS of [Formula: see text], then it is called a DCDS graph. First, a necessary but not sufficient condition is proposed for cubic (3-regular) graph to be a DCDS graph. To be exact, if a cubic graph is a DCDS graph, there are at most four disjoint triangles in it. Next, if a connected graph [Formula: see text] is a DCDS graph, a simple but nontrivial upper bound [Formula: see text] of the girth [Formula: see text] is presented.

scholarly journals Perfect and quasiperfect domination in trees

2016 ◽  
Vol 10 (1) ◽  
pp. 46-64
Author(s):  
José Cáceres ◽  
Carmen Hernando ◽  
Mercè Mora ◽  
Ignacio Pelayo ◽  
María Puertas
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Linear Algorithm ◽  
G 12 ◽  
Graph Parameters

A k??quasiperfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S. The cardinality of a minimum k-quasiperfect dominating set in G is denoted by ?1k(G). These graph parameters were first introduced by Chellali et al. (2013) as a generalization of both the perfect domination number ?11(G) and the domination number ?(G). The study of the so-called quasiperfect domination chain ?11(G) ? ?12(G)?... ? ?1?(G) = ?(G) enable us to analyze how far minimum dominating sets are from being perfect. In this paper, we provide, for any tree T and any positive integer k, a tight upper bound of ?1k(T). We also prove that there are trees satisfying all possible equalities and inequalities in this chain. Finally a linear algorithm for computing ?1k(T) in any tree T is presented.

scholarly journals On autographix conjecture regarding domination number and average eccentricity

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 699-710 ◽  
Author(s):  
Li-Dan Pei ◽  
Xiang-Feng Pan ◽  
Jing Tian ◽  
Gui-Qin Peng
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Mean Value ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Maximal Distance ◽  
The Mean ◽  
Average Eccentricity

The eccentricity of a vertex is the maximal distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. A set S ? V(G) is a dominating set of a graph G if NG(v) ? S ? 0 for any vertex v ? V(G)\S. The domination number (G) of G is the minimum cardinality of all dominating sets of G. In this paper, we correct an AutoGraphiX conjecture regarding the domination number and average eccentricity, and present a proof of the revised conjecture. In addition, we establish an upper bound on ?(T)-ecc(T) for an n-vertex tree T.

scholarly journals Claw-free graphs with equal 2-domination and domination numbers

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2795-2801 ◽  
Author(s):  
Adriana Hansberg ◽  
Bert Randerath ◽  
Lutz Volkmann
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Line Graphs ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination Numbers

For a graph G a subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number k(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number 1(G) is the usual domination number (G). Fink and Jacobson showed in 1985 that the inequality ?k(G)?(G)+k?2 is valid for every connected graph G. In this paper, we concentrate on the case k = 2, where k can be equal to ?, and we characterize all claw-free graphs and all line graphs G with ?(G) = ?2(G).

scholarly journals Dominating Cocoloring of Graphs

2019 ◽  
Vol 9 (1) ◽  
pp. 2545-2547
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Cochromatic Number

A -cocolouring of a graph is a partition of the vertex set into subsets such that each set induces either a clique or an independent set in . The cochromatic number of a graph is the least such that has a -cocolouring of . A set is a dominating set of if for each , there exists a vertex such that is adjacent to . The minimum cardinality of a dominating set in is called the domination number and is denoted by . Combining these two concepts we have introduces two new types of cocoloring viz, dominating cocoloring and -cocoloring. A dominating cocoloring of is a cocoloring of such that atleast one of the sets in the partition is a dominating set. Hence dominating cocoloring is a conditional cocoloring. The dominating co-chromatic number is the smallest cardinality of a dominating cocoloring of .(ie) has a dominating cocoloring with -colors .

scholarly journals On Independent Transversal Dominating Sets in Graphs

2021 ◽  
Vol 14 (1) ◽  
pp. 149-163
Author(s):  
Daven Sapitanan Sevilleno ◽  
Ferdinand P. Jamil
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Positive Integers

A set S ⊆ V (G) is an independent transversal dominating set of a graph G if S is a dominating set of G and intersects every maximum independent set of G. An independent transversal dominating set which is a total dominating set is an independent transversal total dominating set. The minimum cardinality γit(G) (resp. γitt(G)) of an independent transversal dominating set (resp. independent transversal total dominating set) of G is the independent transversal domination number (resp. independent transversal total domination number) of G. In this paper, we show that for every positive integers a and b with 5 ≤ a ≤ b ≤ 2a − 2, there exists a connected graph G for which γit(G) = a and γitt(G) = b. We also study these two concepts in graphs which are the join, corona or composition of graphs.

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