scholarly journals An Improved Nordhaus–Gaddum-Type Theorem for 2-Rainbow Independent Domination Number

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 402
Author(s):  
Enqiang Zhu
Keyword(s):  
Applied Mathematics ◽  
Domination Number ◽  
Type Theorem ◽  
Independent Sets ◽  
Independent Domination ◽  
Integer Labeling ◽  
Vertex Disjoint

For a graph G, its k-rainbow independent domination number, written as γrik(G), is defined as the cardinality of a minimum set consisting of k vertex-disjoint independent sets V1,V2,…,Vk such that every vertex in V0=V(G)\(∪i=1kVi) has a neighbor in Vi for all i∈{1,2,…,k}. This domination invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics and Computation 333(15), 2018: 353–361), which aims to compute the independent domination number of G□Kk (the generalized prism) via studying the problem of integer labeling on G. They proved a Nordhaus–Gaddum-type theorem: 5≤γri2(G)+γri2(G¯)≤n+3 for any n-order graph G with n≥3, in which G¯ denotes the complement of G. This work improves their result and shows that if G≇C5, then 5≤γri2(G)+γri2(G¯)≤n+2.

scholarly journals On the 2-Packing Differential of a Graph

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
A. Cabrera Martínez ◽  
M. L. Puertas ◽  
J. A. Rodríguez-Velázquez
Keyword(s):  
Domination Number ◽  
Type Theorem ◽  
Total Domination Number ◽  
Roman Domination ◽  
Packing Number ◽  
Roman Domination Number ◽  
Product Graphs ◽  
Vertex Set ◽  
Independent Domination ◽  
Graph Parameters

AbstractLet G be a graph of order $${\text {n}}(G)$$ n ( G ) and vertex set V(G). Given a set $$S\subseteq V(G)$$ S ⊆ V ( G ) , we define the external neighbourhood of S as the set $$N_e(S)$$ N e ( S ) of all vertices in $$V(G){\setminus } S$$ V ( G ) \ S having at least one neighbour in S. The differential of S is defined to be $$\partial (S)=|N_e(S)|-|S|$$ ∂ ( S ) = | N e ( S ) | - | S | . In this paper, we introduce the study of the 2-packing differential of a graph, which we define as $$\partial _{2p}(G)=\max \{\partial (S):\, S\subseteq V(G) \text { is a }2\text {-packing}\}.$$ ∂ 2 p ( G ) = max { ∂ ( S ) : S ⊆ V ( G ) is a 2 -packing } . We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $$\partial _{2p}(G)+\mu _{_R}(G)={\text {n}}(G)$$ ∂ 2 p ( G ) + μ R ( G ) = n ( G ) , where $$\mu _{_R}(G)$$ μ R ( G ) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on $$\mu _{_R}(G)$$ μ R ( G ) , and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.

scholarly journals Independence and domination on shogiboard graphs

2017 ◽  
Vol 4 (8) ◽  
pp. 25-37 ◽  
Author(s):  
Doug Chatham
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
The Other ◽  
Independent Domination

Abstract Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals Independent Domination Stable Trees and Unicyclic Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 820
Author(s):  
Pu Wu ◽  
Huiqin Jiang ◽  
Sakineh Nazari-Moghaddam ◽  
Seyed Mahmoud Sheikholeslami ◽  
Zehui Shao ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Basic Properties ◽  
Independent Domination ◽  
Independent Dominating Set

A set S ⊆ V ( G ) in a graph G is a dominating set if every vertex of G is either in S or adjacent to a vertex of S . A dominating set S is independent if any pair of vertices in S is not adjacent. The minimum cardinality of an independent dominating set on a graph G is called the independent domination number i ( G ) . A graph G is independent domination stable if the independent domination number of G remains unchanged under the removal of any vertex. In this paper, we study the basic properties of independent domination stable graphs, and we characterize all independent domination stable trees and unicyclic graphs. In addition, we establish bounds on the order of independent domination stable trees.

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.

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