EXPLORING AT MOST TWIN OUTER PERFECT DOMINATION NUMBER FOR GENERAL BINARY TREE AND SOME SPECIAL GRAPHS

A set S⊆ V(G) said to be an at most twin outer perfect dominating set if for every vertex v∈V-S, l⩽|N(v)∩S|⩽2 and < V-S> has at least one perfect matching. The minimum cardinality of at most twin outer perfect dominating set is called the at most twin outer perfect domination number and γ atop (G) denotes this number. This was initiated by G.Mahadevan.et.al., recently. Here we find this number for general Binary-tree. corona product of paths and cycles and lotus inside graph.


Motivation
A set V S  is called a complementary perfect dominating set, if S is a dominating set of G and the induced sub graph > < S V  has a perfect matching. The minimum cardinality taken over all complementary perfect dominating sets is called the complementary perfect domination number and is denoted by ) (G cp  this was initiated by Paulraj Joseph et.al., [7]. A subset  k for every vertex, for any non-negative integer j and k. this was first studied by Mustapha Chellali et.al., [6]. Later,in [8], Xiaojing Yang and Baoyindureng Wu, extended this study and initiated [1,2]-domination number. A vertex set S in graph G is [1,2] is adjacent to either one or two vertices in S. The minimum cardinality of a [1,2]-set of G is denoted by ) ( [1,2] G  and it is said to be [1,2]domination number of G. The above paper, stimulates us to do something using the condition of perfect matching in the complement of [1,2] dominating set and we call this parameter as atmost twin outer perfect domination number of a graph. Motivated by above in [3], G.Mahadevan, et.al., introduced the concept of at most twin outer perfect domination number of a graph. In [4], the authors obtained many results of this parameter with graph colouring.

2.Introduction
Graphs considered here are simple and connected.The graph lotus inside circle is denoted by t LIC 3  t and is defined as follows. Let t S be the star graph with vertices whose center is 0 x . Let t C be the cycle of length t whose vertices are we join t y with 1 x and t x . The corona G 1 G 2 is obtained by taking one copy of G 1 of order p 1 and p 1 copies of G 2 and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2 .A tree is a 2-array tree (binary tree) if every vertex has degreee at most 3. If a root is appointed in the tree, then every vertex has atmost 2 children. Throughout the paper, we find this number for lotus inside graph,corona product of two graphs and 2-array tree.
is an atopd-set.

Thus ,
As D is an atopd set implies that is even, if there exists an atopd set T such that |T| |D|, then will be odd implies that does not have a perfect matching. Hence . Therefore Proof Let |V(P r )|and |V(P s )| be r and s respectively. Let P r P s be the corona product of P r and P s . Here V(P r P s )={u i ,v ij / 1 } Case is is even Consider the set S ={u 1 , u 2 ,…u r } be any atopd-set. Since, every vertex is linked with one vertex in S, this gives that minimum atopd-set in P r P s is S. Here s is even has a perfect matching. Hence S is an atopd-set in P r P s . This gives = r If S contains atleast a vertex not on P r , then atopd-set is not possible, hence . Case ii s is odd Let S = {u 1 , u 2 ,…u r , v 11 , v 21 ,….v s1 } be any atopd-set. Since,every vertex is linked with one vertex in S, this gives that minimum atopd-set in P r P s is S and also has a perfect matching. Hence S is an atopd-set in P r P s . This gives = 2r IfS contains all the vertices only on the path P r , then has no perfect matching. Hence S contains r times of vertices in the path P s in addition to all the vertices in the path P r hence . Hence the result.    Let A 0 = 1 |A 0 |=1 Let is an atopd-set of B 3r+1 |S|=|A 0 |+|A 1 |+|A 2 |+….+|A 3r+1 | = 1+2+2 2 +2 3 +……+2 3r-7 +2 3r-6 +2 3r-5 +2 3r-3 +2 3r-2 +2 3r-1 +2 3r +2 3r+1 = (1+2+2 2 +2 3 +……..

Illustration
Consider the level B 5 The dark dot denotes atopd-set and let it be denoted by S then|S|=45.

5.Conclusion
In this paper we explored the general result for atmost twin outer perfect domination number for 2array tree, Lotus inside the flower and corona products of paths and cycles. The authors obtained this atmost twin outer perfect domiation number for different various types products of graphs which will be reported in the subsequent papers.