Planar graph characterization of NDSS graphs

Yamuna, M.; Elakkiya, A.

doi:10.1088/1757-899x/263/4/042129

Cited by 6 publications

(5 citation statements)

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“…In 2014, a characterization of planar graphs when G and G are  -stable graphs were provided in [43].…”

Section: Theorem[42]mentioning

confidence: 99%

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

Yamuna¹,

Karthika²

2016

IJET

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Abstract-A dominating set of G is a set of vertices of G such that every vertex in V -D is adjacent to a vertex in D.The domination number of G, denoted by  ( G ), is the minimum cardinality of a dominating set. Different kinds of graph operations have different outcomes on the domination number of a graph. In this paper we present a brief survey on the impact of different kinds of graph operation on the domination number of a graph.Keyword -Domination, Domination Number, Graph Operation. I. INTRODUCTIONThe study of domination number in graph theory was introduced by Claude Berge in 1958, on his book "Theory of Graphs and its Applications" [1]. But domination number was studied in the name of coefficient of external stability. At the first time the name dominating set and domination number was used by Oyestein Ore in [2]. At the beginning stage the domination number was denoted by d ( G ). Later, in 1977, the notation  ( G ) is used to denote the domination number in Since the evolution of domination theory, types of domination are also established and studied by many researchers. But the original dominating set is the base for all types of domination. Different kinds of graph operations change the graphs and hence influence the domination number of the graph also. In this survey we concentrate on the effect of various kinds of graph operations on the domination number. There are several results relating graph operations and different types of domination. But we restrict our survey only to graph operations and dominating sets. Several results are available in this regard, and many have been omitted in this brief survey. We apologize to the authors for the omission. Many results omitted here can be found in [4][5] and in the articles in the reference section. This survey is restricted to presentation of possible results that describe the effects of the graph operations on the domination number of a graph. The connectivity k = k ( G ) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. A set of independent edges in a graph G is called a matching of G. A 1 -factor or a perfect matching of a graph is a partition of its vertices into adjacent pairs. II. GRAPH THEORY TERMINOLOGY AND CONCEPTSP n , C n , K n , denotes the path, cycle and complete graph with n vertices respectively. The wheel graph is denoted by W n , and it is defined as K 1 + C n -1 . We denote the radius and diameter of G by rad( G ) and diam ( G ), respectively. The minimum and maximum degree on the vertices of G is denoted by  ( G ) and  ( G ) respectively. If  ( G ) =  ( G ) = r, then all points have the same degree and G is called regular of degree r. The regular graphs are those of degree 3, such graphs are called cubic. The complement G of a graph G also has V ( G ) as its vertex set, but two vertices are adjacent in G if and only if they are not adjacent in G.

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“…In 2014, a characterization of planar graphs when G and G are  -stable graphs were provided in [43].…”

Section: Theorem[42]mentioning

confidence: 99%

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

Yamuna¹,

Karthika²

2016

IJET

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“…In [ 5 ], M. Yamuna et al have characterized the planarity and outer planarity of complement of NDSS graphs. In [6], M. Yamuna et al have determined the domination number of G  , G  , chromatic polynomial of G  , spanning tree of G  , number of spanning trees of G  from G.…”

Section: Introductionmentioning

confidence: 99%

Chromatic Polynomial of Domination Subdivision Non Stable Graphs

Elakkiya¹,

Yamuna²

2019

IJITEE

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“…In [1] Bing Zhou investigated the dominating --color number, d  (G) , of a graph G. In [2], [3], M. Yamuna et al introduced uniquely colorable graphs and also provided the constructive characterization of  -uniquely colorable trees and characterized planarity of complement of  -uniquely colorable graphs. In [4], [5],M. Yamuna et al introduced Non domination subdivision stable graphs (NDSS) andcharacterized planarity of complement of NDSS graphs…”

Section: Introductionmentioning

confidence: 99%

Planar and Non Planar Construction of - Uniquely Colorable Graph

Elakkiya

Yamuna

2018

IJET

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A uniquely colorable graph G whose chromatic partition contains atleast one g - set is termed as a g - uniquely colorable graph. In this paper, we provide necessary and sufficient condition for and G* to be g - uniquely colorable whenever G g- uniquely colorable and also provide constructive characterization to show that whenever G is g- uniquely colorable such that |P | ³ 2, G can be both planarand non planar.

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Planar graph characterization of NDSS graphs

Cited by 6 publications

References 1 publication

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

A Survey on the Effect of Graph Operations on the Domination Number of a Graph

Chromatic Polynomial of Domination Subdivision Non Stable Graphs

Planar and Non Planar Construction of - Uniquely Colorable Graph

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