scholarly journals Upper paired domination versus upper domination

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Graph Classes ◽  
Paired Domination Number ◽  
Upper Domination Number ◽  
Paired Domination ◽  
Cactus Graphs ◽  
Open Question

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

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].

scholarly journals Bounding the paired-domination number of a tree in terms of its annihilation number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 523-529 ◽  
Author(s):  
Nasrin Dehgardi ◽  
Seyed Sheikholeslami ◽  
Abdollah Khodkar
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Degree Sequence ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Paired Domination Number ◽  
Isolated Vertex ◽  
Paired Domination

A paired-dominating set of a graph G=(V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by ?pr(G), is the minimum cardinality of a paired-dominating set of G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we prove that for any tree T of order n?2,?pr(T)? 4a(T)+2/3 and we characterize the trees achieving this bound.

scholarly journals Γ -Paired dominating graphs of some paths

2018 ◽  
Vol 189 ◽  
pp. 03029
Author(s):  
Pannawat Eakawinrujee ◽  
Nantapath Trakultraipruk
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
The Other ◽  
Maximum Cardinality ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination

A paired dominating set of a graph G = (V(G),E(G)) is a set D of vertices of G such that every vertex is adjacent to some vertex in D, and the subgraph of G induced by D contains a perfect matching. The upper paired domination number of G, denoted by Γpr(G) is the maximum cardinality of a minimal paired dominating set of G. A paired dominatin set of cardinality Γ pr(G) is called a Γpr(G) -set. The Γ -paired dominating graph of G, denoted by ΓPD(G), is the graph whose vertex set is the set of all Γ pr(G) -sets, and two Γpr(G) -sets are adjacentin ΓPD(G) if one can be obtained from the other by removing one vertex and adding another vertex of G. In this paper, we present the Γ-paired dominating graphs of some paths.

scholarly journals On the Paired-Domination Subdivision Number of Trees

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1135
Author(s):  
Shouliu Wei ◽  
Guoliang Hao ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar ◽  
Hossein Karami
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Paired Domination Number ◽  
Minimum Number ◽  
Paired Domination ◽  
Domination Subdivision Number

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γpr(G) of G. The paired-domination subdivision number sdγpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T≠P5 of order n≥3 and each edge e∉E(T), sdγpr(T)+sdγpr(T+e)≤n+2.

scholarly journals Almost well-dominated bipartite graphs with minimum degree at least two

2020 ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Upper Domination Number ◽  
The Difference ◽  
Two Parameters

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

scholarly journals A Note on Total and Paired Domination of Cartesian Product Graphs

10.37236/2535 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
K. Choudhary ◽  
S. Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Dominating Set ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Paired Domination ◽  
Product Of Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

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