An Investigation of Corona Domination Number for Some Special Graphs and Jahangir Graph

: In this work, the study of corona domination in graphs is carried over which was initially proposed by G. Mahadevan et al. Let 𝐻 be a simple graph. A dominating set S of a graph is said to be a corona-dominating set if every vertex in < 𝑆 > is either a pendant vertex or a support vertices. The minimum cardinality among all corona-dominating sets is called the corona-domination number and is denoted by 𝛾 𝐶𝐷 (𝐻). (i.e) 𝛾 𝐶𝐷 (𝐻) = 𝑚𝑖𝑛{|𝑆| ∶ 𝑆 𝑖𝑠 𝑎 𝐶𝐷 − 𝑠𝑒𝑡 𝑜𝑓 𝐻} . In this work, the exact value of the corona domination number for some specific types of graphs are given. Also, some results on the corona domination number for some classes of graphs are obtained and the method used in this paper is a well-known number theory concept with some modification this method can also be applied to obtain the results on other domination parameters.


Introduction:
Every graph = ( ( ), ( )) considered here are connected finite, undirected, without isolated vertex and loops. A dominating set 1 is a set of vertices of with the condition that every ∈ − , ( , ) = 1. The minimum cardinality among all the dominating sets is called the domination number of , denoted by ( ). The concept of corona domination was introduced by G.Mahadevanet al. 2

. The corona domination number (CD number)
is a minimum cardinality of the dominating set , with the subgraph induced by having either pendant or support vertices only. For example see Fig. 1. In recent years many authors have studied the different concept in graph theory and domination theory such as order sum graph 3 , tadpole domination 4 etc. Let ′( ) be the middle graph of if two vertices x and y in the vertex set of ′( ) are adjacent if , are in ( ) and , are adjacent in is in ( ), is in ( ) and is incident to in . The central graph ′( ) of is obtained by subdividing each edge in ( ) and joining all the non-adjacent vertices in .
A wheel graph 5 1, , ≥ 3 is obtained by joining a single vertex to all the vertices of a cycle . A graph obtained by attaching a pendant edge at each vertex of is called helm graph 5 .Joining the pendant vertices of the helm graph to form a cycle will give a closed helm graph . The friendship graph 5 is obtained by attaching the -copies of 3 at a common vertex. A graph constructed by joining a to an end vertex of by a bridge is called the tadpole graph , . The Cartesian product of two paths 2 and gives the ladder graph 5 2 ⊠ . A single vertex is adjacent to vertices of at a distance r to one another on is called a Jahangir graph , . Consider a sequence of cycle 4 say 4 1 , 4 2 , 4 3 ,…, 4 , a diamond snake graph is obtained by pasting 1 −1 where 1 ≤ ≤ . The ℎ power of the graph is graph with same set of vertices and two vertices and in are adjacent whenever ( , ) ≤ . The shadow graph ' ( ) of a graph is that a graph obtained by adding a new vertex ' for each vertex of and joining ′ to the neighbors of in .
The CD-number for the fan and generalized fan graph , ,the complement of the ladder graph for m-shadow and m-splitting graph of and is same as the CD-number of and , the CDnumber for the Moster spindle graph 5 , the CDnumber for Wagner graph is 3, the CD-number for King's tour graph is 12. Examining the CD-number for some special graphs Theorem 1: 6 Let be a diamond snake graph. Then ( ) = + 1 Theorem 2: Let be a diamond snake graph. Then ( ) = + 1 Proof: Let ( ) = { 1 , 2 , . . . , , 1 , 2 , . . . , −1 , 1 , 2 , . . . , −1 }then

Proof:Let
Therefore the proof.
Therefore the proof.
Therefore the proof.

Conclusion:
In this work, the CD-number for some special graphs and the Jahangir graph are found. Moreover, these results are characterized with other domination parmaeters, which will be reported in the successive papers.