Independence Numbers in Trees

The independence number ( ) G α of a graph G is the maximum cardinality among all independent sets of G. For any tree T of order n ≥ 2, it is easy to see that ( )       n T n α 1 2 ≤ ≤ − . In addition, if there are duplicated leaves in a tree, then these duplicated leaves are all lying in every maximum independent set. In this paper, we will show that if T is a tree of order n ≥ 4 without duplicated leaves, then ( )       n T α 2 1 3 − ≤ . Moreover, we constructively characterize the extremal trees T of order n ≥ 4, which are without duplicated leaves, achieving these upper bounds.


Introduction
All graphs considered in this paper are finite, loopless, and without multiple edges.For a graph G, we refer to ( ) V G and ( ) E G as the vertex set and the edge set, respectively.The cardinality of ( )

{ }
: and − by removing all vertices in A and all edges incident to these vertices and the complement of A is the set . For notation and terminology in graphs we follow [1] in general.
A set ( ) The independence problem is to find an α -set in G.The problem is known to be NP-hard in many special classes of graphs.Over the past few years, several studies have been made on [2]- [6]).For any tree T of order 2 n ≥ , it is easy to see that ( ) . In addition, if there are duplicated leaves in a tree, then these duplicated leaves are all lying in every maximum independent set.In this paper, we will show that if T is a tree of order 4 n ≥ without duplicated leaves, then ( ) . Moreover, we constructively characterize the extremal trees T of order 4 n ≥ , which are without duplicated leaves, achieving these upper bounds.

The Upper Bound
In this section, we will show a sharp upper bound on the independence number of a tree T without duplicated leaves.

Lemma 1 If H is an induced subgraph of G, then ( ) ( )
Proof.If S is an α -set of H, then S is an independent set of G.It follows that ( ) ( ) ) If T is a tree of order 3 n ≥ , then there exists an L α -set of T.

Lemma 4 For an integer 4
n ≥ , ( ) Proof.It is straightforward to check that 2 1 2 3  : Since P n is a tree of order 4 n ≥ , by Lemma 2, we have that ( ) . Suppose that there exists an independent set I of , then there exists i, . This is a contradiction, therefore we obtain that ( ) Theorem 1 If T is a tree of order 4 n ≥ without duplicated leaves, then ( ) Proof.We prove it by induction on 4 n ≥ .By Lemma 4 and T is a tree without duplicated leaves, it's true for all 6 n ≤ .For all 7 n ≥ we assume that the assertion is true for all n n ′ < .Suppose that T is a tree of order 7 n ≥ without duplicated leaves and x is a leaf lying on a longest path of T. Let ( ) . Since T has no duplicated leaves, this implies that , then T' is a tree of order 2 n − .For the case in which T' has no duplicated leaves, by induction hypothesis, we have that ( ) ( ) . Since an α -set of T', together with { } x , form an α -set of T. Therefore we obtain that ( ) ( ) . For the other case in which T' has duplicated leaves z and z′ , then is a tree of order 3 4 n − ≥ without duplicated leaves.By induction hy- pothesis, we have that ( ) ( ) an α -set of T. Therefore, we obtain that ( ) ( ) . Hence we conclude that ( ) Note that the result in Theorem 1 is sharp and some such T are illustrated below.

Extremal Trees
Let ( ) be the class of all trees T of order 4 n ≥ without duplicated leaves such that ( ) We will constructively characterize these extremal trees.Let ( ) ( ) U T , respectively, denote the collections of all leaves and all support vertices of T. First, we define four operations on a tree T of order 4 n ≥ as follows, where , where 0,1 T n = ≡ (mod 3).Operation O3.Join a vertex u of T to a leaf 2 v of 2 P (say 2 1 2 : , where 1, 2 , Proof.It's true for all 6 n ≤ .So we assume that 7 n ≥ .Since I is an L α -set of T, this implies that ( ) c U T I ⊆ .By Theorem 1, we have that ( ) ( ) ( ) , these imply that ( ) be a tree of order ( ) with an L α -set I. Suppose that T' is obtained from T by Operation O1, then ( ) is a tree of order is a tree of order 2 n ≡ (mod 3) with an L α -set I, by Lemma 5, then ( ) Let T' be the tree obtained from T by Operation O1.Since u I ∈ , this implies that u is not a support vertex of T and T' is a tree of order 1 n + without duplicated leaves.On the other hand, 1 O I is an independent set of T ′ with ( ) , where 2 n ≡ (mod 3).Hence ( ) ( ) . In conclusion, ( ) be a tree of order ( ) is a tree of order 1 n + and Proof.Note that such a tree T exists, as, for instance, the tree in Figure 1 is as desired.If is a tree of order 0,1 n ≡ (mod 3) with an L α -set I such that ( ) Let T' be the tree obtained from T by Operation O2.Since u is not a support vertex of T, this implies that T' is a tree of order 1 n + without dupli- cated leaves.And , where 0,1 n ≡ (mod 3).Hence ( ) ( ) . In conclusion, ( ) is a tree of order 1 n + with an L α -set I O2 .
 Lemma 8 Let ( ) be a tree of order ( ) Operation O3, then ( ) is a tree of order 2 n + and I O3 is an L α -set of T'.
Proof.Note that T' is a tree of order n + 2 without duplicated leaves.And I O3 is an independent set of T' with ( ) , where 1, 2 n ≡ (mod 3).Hence ( ) ( ) . In conclusion, ( ) is a tree of order 2 n + with an L α - set I O3 . Lemma 9 Let ( ) be a tree of order 4 n ≥ with an L α -set I.If T' is obtained from T by Operation O4, then ( ) is a tree of order 3 n + and I O4 is an L α -set of T'.
Proof.Note that T' is a tree of order 3 n + without duplicated leaves.And I O4 is an independent set of T' with ( ) . Hence ( ) ( ) . In conclusion, ( ) is a tree of order 3 n + with an L α -set I O4 .Proof.If T is in C , by Lemmas 6, 7, 8 and 9, then T is in T .Now, we want to show the converse by con- tradiction.Suppose to the contrary that there exists a tree T ∈T and T ∉C such that T is as small as possible.We can see that 7 is an independent set of T', this implies that ( ) ( ) Theorem 1, we have that and 0,1 n ≡ (mod 3).This follows that ( ) , where .Then T ′′ is a tree of order 3 n − .Since z' is a leaf of T, this implies that z and z' are in every L α -set of T. For an L α -set I of T, { } . Then ( ) ( ) . This follows that ( ) , where 3 n n ′′ = − , by hypothesis, T ′′ ∈C .Note that T can be obtained from T ′′ by Operation O4, this implies that T ∈C , which is a contradiction.By Cases 1 and 2, we conclude that T is in T , then T is in C . Now, we obtain the main theorem in this paper.
Theorem 3 Suppose that T is a tree of order 4 n ≥ without duplicated leaves, then ( ) . Furthermore, the equality holds if and only if T ∈C .

α
of a graph G is the maximum cardinality among all independent sets of G.For any tree T of order n ≥ 2, it is easy to see that in a tree, then these duplicated leaves are all lying in every maximum independent set.In this paper, we will show that if T is a tree of order n ≥ 4 without duplicated leaves,

1 G
G is called the order of G, denoted by G .The (open) neighborhood ( ) G N x of a vertex x is the set of vertices adjacent to x in G, and the close neighborhood x is said to be a leaf if ( )N x = .A vertex v of G is a support vertex if it is adjacent to a leaf in G.Two distinct vertices u and v are called duplicated if u and v are duplicated vertices in a tree, and then they are both leaves.The n-path n P is the path of order 1 n ≥ .For a subset ( ) A V G ⊆ , the induced subgraph induced by A is the graph G A with vertex set A and the edge set ( ) ( ) Let

Let C be the class of all trees obtained from 4 P or 5 P
by a finite sequence of Operations O1-O4.Suppose that Theorem 2 T is in C if and only if T is in T .

T
≥ .Let : P x y z − − − be a longest path of T. Then .We consider two cases.Case 1. T' has no duplicated leaves.For an L α -set I of T, T ′ ∈C .Note that T can be obtained from T ′ by Operation O3, this implies that T ∈C , which is a Case 2. has duplicated leaves z and z'.