Nonsplit Neighbourhood Tree Domination Number In Connected Graphs

: Let G = (V, E) be a connected graph. A subset D of V is called a dominating set of G if N[D] = V. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by  (G). A dominating set D of a graph G is called a tree dominating set (tr - set) if the induced subgraph  D  is a tree. The tree domination number γ tr (G) of G is the minimum cardinality of a tree dominating set. A tree dominating set D of a graph G is called a neighbourhood tree dominating set (ntr - set) if the induced subgraph  N(D)  is a tree. The neighbourhood tree domination number γ ntr (G) of G is the minimum cardinality of a neighbourhood tree dominating set. A tree dominating set D of a graph G is called a nonsplit tree dominating set (nstd - set) if the induced subgraph  V  D  is connected. The nonsplit tree domination number γ nstd (G) of G is the minimum cardinality of a nonsplit tree dominating set. A neighbourhood tree dominating set D of G is called a nonsplit neighbourhood tree dominating set, if the induced subgraph  V(G) ‒ D  is connected. The nonsplit neighbourhood tree domination number γ nsntr (G) of G is the minimum cardinality of a nonsplit neighbourhood tree dominating set of G. In this paper, bounds for γ nsntr (G) and its exact values for some particular classes of graphs and cartesian product of some standard graphs are found.


INTRODUCTION
The graphs considered here are nontrivial, finite and undirected.The order and size of G are denoted by n and m respectively.If D  V, then of G which are adjacent to v. The concept of domination in graphs was introduced by Ore [13].A subset D of V is called a dominating set of G if N[D] = V.The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by (G).Xuegang Chen, Liang Sun and Alice McRac [14] introduced the concept of tree domination in graphs.A dominating set D of G is called a tree dominating set, if the induced subgraph D is a tree.The minimum cardinality of a tree dominating set of G is called the tree domination number of G and is denoted by tr(G).Kulli and Janakiram [8,9] introduced the concept of split and nonsplit domination in graphs.
A dominating set D of a graph G is called a nonsplit dominating set if the induced subgraph V  D is connected.The nonsplit domination number γnsd(G) of G is the minimum cardinality of a nonsplit dominating set.Muthammai and Chitiravalli [11,12] defined the concept of split and nonsplit tree domination in graphs.A tree dominating set D of a graph G is called a nonsplit tree dominating set if the induced subgraph V  D is connected.The nonsplit tree domination number γnstd(G) of G is the minimum cardinality of a nonsplit tree idominating set.
V.R. Kulli introduced the concepts of split and nonsplit neighbourhood connected domination in graph.A neighbourhood dominating set D of a graph G is called a nonsplit neighbourhood dominating set if the induced subgraph V  D is connected.The nonsplit neighbourhood domination number γnsntd(G) of G is the minimum cardinality of a nonsplit neighbourhood dominating set.
In this paper, bounds for γnsntr(G) and its exact values for some particular classes of graphs and cartesian product of some standard graphs are found.

MAIN RESULTS
In this section, nonsplit neighbourhood tree domination number is defined and studied.

Nonsplit Neighbourhood Tree Domination Number in Connected Graphs Definition 3.1.1:
A neighbourhood tree dominating set D of G is called a nonsplit neighbourhood tree dominating set, if the induced subgraph V(G) -D is connected.The nonsplit neighbourhood tree domination number γnsntr(G) of G is the minimum cardinality of a nonsplit neighbourhood tree dominating set of G.
Not all connected graphs have a nonsplit neighbourhood tree dominating set.For example, the Path Pn(n > 5) has a neighbourhood tree dominating set, but no nonsplit neighbourhood tree dominating set.
If the nonsplit neighbourhood tree domination number does not exist for a given connected graph G, then nsntr(G) is defined to be zero.Every nonsplit neighbourhood tree dominating set is a dominating set and also a neighbourhood tree dominating set.Therefore, γ(G) ≤ γntr(G) ≤ γnsntr(G).Therefore, for any nontrivial connected graph G, γntr(G) = min{γsntr(G), γnsntr(G)}.
In the following, the exact values of γsntr(G) for some standard graphs are given.If T is a tree which is not a star, then nsntr(T) ≤ n -2.

Proof:
Suppose T is not a star.Then T has two adjacent cut vertices u and v, such that deg u, deg v ≥ 2. This implies that D = {V -{u, v}} is a nonsplit neighbourhood tree dominating set of T. Therefore, nsntr(T) ≤ │D│= │V(T) -{u, v} │= n -2.

Nonsplit Neighbourhood Tree Domination Number of Cartesian product of Graphs
In this section, nonsplit neighbourhood tree domination numbers of P2  Cn, P3  Cn, P2  Pn, P3  Pn are found.

Let
Here, v2i,2 is adjacent to v2i,3 (i ≥ 1) and v2i-1,2 is adjacent to v2i-1,1 (i ≥ 1).Therefore, D is a dominating set of G and N(D)  Pn ⃘ P1.Since N(D) is a tree and V -D is connected, D is a nonsplit neighbourhood tree dominating set of G and is minimum.
Let D = {v31, v2,2}.Then D  V(G).Here, v11, v21 are adjacent to v31 and v12, v32 are adjacent to v2,2.Therefore, D is a dominating set of G and N(D)  P4.Since N(D) is a tree and V(G) -D is connected, D is a nonsplit neighbourhood tree dominating set of G and ntr(G)  D= 2.

  
and {v1j, v2j, ... ,vnj}  Cn j , j = 1, 2, ... ,n where P3 i is the i th copy of P3 and Cn j is the j th copy of Cn in G.
Let D = {v31, v12, v33}.Then D  V(G).Here, v22 is adjacent to v12 and v11, v21, v32 are adjacent to v31 and v32, v13, v23 are adjacent to v33.Therefore, D is a dominating set of G and N(D) is a connected graph obtained from P5 by attaching a pendant edge at v22.Since N(D) is a tree and V(G) -D is connected, D is a nonsplit neighbourhood tree dominating set of G and nsntr(G)  D= 3.

  
vD N(D) = N(v)  and N[D] = N(D)  D where N(v) is the set of vertices

Figure 3
Figure 3.4 (a) For any path Pn on n vertices, γnsntr(Pn) = n -2, n  4. (b) If G is a spider, then nsntr(G) = n + 1. (c) If G is a wounded spider, then nsntr(G) = p + 1, where p is the number of pendant vertices which are adjacent to nonwounded legs.(d) For any triangular cactus graph Tp whose blocks are p triangles with p ≥ 1, nsntr(Tp) = p where p > 2 and p is odd.(e) If Sm,n, (1 ≤ m ≤ n) is a double star, then nsntr(Sm,n) = m + n.Theorem 3.1.1:

2 P
Here, v11 and v22 are adjacent to v12 and vn1 and vn-1,2 are adjacent to vn2 and v2i+1,2 is adjacent to v2i+1,1 (i ≥ 1).Therefore, D is a dominating set of G and N(D)  3n -1 .Since N(D) is a tree and V(G) -D is connected, D is a nonsplit neighbourhood tree dominating set of G and is minimum.Hence nsntr(G) = D= nsntr(P2  P3) = 2, the set {v31, v12} is a minimum nonsplit neighbourhood tree dominating set of P2  Pn , where v21, v22 are the vertices of degree 3 in P2  P3.