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.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

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 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 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.

A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450025 ◽  
Author(s):  
XIUMEI WANG ◽  
WEIPING SHANG ◽  
YIXUN LIN ◽  
MARCELO H. CARVALHO
Keyword(s):  
Convex Hull ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Matching Polytopes ◽  
Matching Polytope

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.

Clique doubly connected domination in the join and lexicographic product of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050066
Author(s):  
Enrico L. Enriquez ◽  
Albert D. Ngujo
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Connected Dominating Set ◽  
Lexicographic Product ◽  
Connected Domination Number ◽  
Vertex Set ◽  
Product Of Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

scholarly journals INTERSECTION GRAPH OF A MODULE

2013 ◽  
Vol 12 (05) ◽  
pp. 1250218 ◽  
Author(s):  
ERGÜN YARANERI
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Undirected Graph ◽  
Domination Number ◽  
Intersection Graph ◽  
Combinatorial Properties ◽  
Algebraic Properties ◽  
Vertex Set ◽  
Multiple Edges

Let V be a left R-module where R is a (not necessarily commutative) ring with unit. The intersection graph [Formula: see text] of proper R-submodules of V is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper R-submodules of V, and there is an edge between two distinct vertices U and W if and only if U ∩ W ≠ 0. We study these graphs to relate the combinatorial properties of [Formula: see text] to the algebraic properties of the R-module V. We study connectedness, domination, finiteness, coloring, and planarity for [Formula: see text]. For instance, we find the domination number of [Formula: see text]. We also find the chromatic number of [Formula: see text] in some cases. Furthermore, we study cycles in [Formula: see text], and complete subgraphs in [Formula: see text] determining the structure of V for which [Formula: see text] is planar.

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 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.

Characterization of some classes of graphs with equal domination number and isolate domination number

2020 ◽  
Vol 12 (05) ◽  
pp. 2050065
Author(s):  
Davood Bakhshesh
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Interval Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Proper Interval Graph ◽  
Domination In Graphs

Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232–241].

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