Set-Valued Graphs

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Available online at www.ispacs.com/jfsva Volume 2012, Year 2012 Article ID jfsva-00127, 17 pages

doi:10.5899/2012/jfsva-00127 Research Article

Set-Valued Graphs

Kumar Abhishek1_∗

, Germina K. Agustine2

(1)Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641 105, India. (2)Mary Matha Arts&Science College, Mananthavady - 670 645, India.

Copyright 2012 c⃝Kumar Abhishek and Germina K. A. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A set-indexer [3] of a given graph G = (V, E) is an assignment f of distinct nonempty subsets of a finite nonempty ‘ground set’Xof cardinalitynto the vertices ofGso that the valuesf⊕

(e), e=uv∈E,obtained as the symmetric differencesf(u)⊕f(v) of the subsets

f(u) andf(v) of X, are all distinct. A set-indexerf of a graph G, is called a segregation

of X on Gif the sets f(V(G)) ={f(u) :u∈V(G)} and f⊕

(E(G)) ={f⊕

(e) :e∈E(G)}

are disjoint, and if, in addition, their union is the set Y(X) = P(X)− {∅} of all the nonempty subsets of X where P(X) denotes the power set of X, then f is called a set-sequential labelingofG.A graph is hence called set-sequentialif it admits a set-sequential labeling with respect to some ‘ground set’ X.A set-indexerf of a (p, q)-graphG= (V, E) is called a set-graceful labeling [3] ofG if there exists nonempty ground set X such that

f⊕₍_E_{) =}_P₍_X₎_{− {∅}}_and _G _is_set-graceful_{if it admits a set-graceful labeling. In this} re-port we provide a complete characterization of set-sequential caterpillar of diameter four. We also a provide a new necessary condition for a graph to be set-sequential.

Keywords: Graphs, Set-indexer, Set-sequential labeling, Set-graceful labeling.

1 Introduction

For all terminology and notation in graph theory, not defined specifically in this paper, we refer the reader to Harary [9]. Unless mentioned otherwise, all the graphs considered in this note are simple, loop-free and finite.

Acharya introduced to the notion of set-valuations of graphs [3], as follows: A set-indexer of a given graph G= (V, E) is an assignmentf of distinct nonempty subsets of a

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finite nonempty ‘ground set’ X={x1, x2, . . . , xn} of cardinality nto the vertices of Gso

that the values f⊕

(e), e=uv ∈E,obtained as the symmetric differences f(u)⊕f(v) of the subsetsf(u) andf(v) ofX,are all distinct. In [3], it is proved that every graph admits a set-indexer. In the recent literature the notion of set-valuation has been surveyed by Hegde [15] using the termset-coloring. However we would follow the terminology used by Acharya in [3]. Acharya and Hegde [6] viewed the notion of set-sequential graphsas a set analogue of the well known sequential graphs, which was independently introduced and studied by Acharya [4] and Slater [13], as follows: A set-indexerf of a graphG,is called a

segregationofX on Gif the setsf(V(G)) ={f(u) :u∈V(G)} andf⊕

(E(G)) ={f⊕ (e) :

e∈E(G)} are disjoint, and if, in addition, their union is the set Y(X) = P(X)− {∅} of all the nonempty subsets of X then f is called a set-sequential labeling[3] of G. A graph is hence called set-sequential if it admits a set-sequential labeling with respect to some ‘ground set’ X. Acharya [3] viewed the notion of set-graceful graphs as a set analogue of the well known graceful graphs, which was introduced by Rosa [2], as follows: A set-indexer(set coloring [15])f of a (p, q)-graphG= (V, E) is called aset-graceful labeling [3] of G if there exists nonempty ground set X such that f⊕

(E) = P(X)− {∅} and G is

set-gracefulif it admits a set-graceful labeling.

In [6], it has been shown that for any (p, q)-graphG,the condition that p+q= 2n₋₁

for some positive integer n is necessary for G to admit a set-sequential labeling and the condition that q = 2n₋_{1 for some positive integer} _n _{is necessary for} _G _{to admit a}

set-graceful labeling; we will call these condition the ‘Acharya-Hegde necessary conditions’. It has also been observed that, in general, this necessary condition for a (p, q)-graph to admit a set-sequential labeling is not sufficient as, for example,P4,the path of order 4,satisfies this necessary condition but the graph is not set-sequential. Further, the following two conjectures were respectively made in [6] and [15].

Conjecture 1.1. [6] No path Pn, n >2 is set-sequential.

Conjecture 1.2. [15] The complete bipartite graphKa,b is set-graceful if and only if it

is a star with a= 1 and b= 2n−1_.

The authors have learnt that Conjecture 1.1 has been disproved [1] and Conjecture 1.2 has been recently settled by Balister, et.al,[16] .

The following is a useful link between the notion of set-sequential and set-graceful graphs.

Theorem 1.1. [3] A graph G is set-sequential if and only ifG+K1 withV(K1) ={v}

has a set-graceful labeling f such that f(v) =∅.

Following Theorems were respectively proved in [7] and [15].

Theorem 1.2. Almost all graphs are not set-graceful.

Theorem 1.3. If G(p >2)has:

1. exactly one or two vertices of even degree or

2. exactly three vertices of even degree at least two of which are adjacent or

3. exactly four vertices of even degree, say v1, v2, v3, v4 such that v1v2 and v3v4 are

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Owing to the Acharya-Hegde necessary conditions, not every graph admits a set-sequential (set-graceful) labeling and in view of Theorem 1.1 and Theorem 1.2 its is quite apparent that “Almost all graphs are not set-sequential”. So far there is no characteri-zation of set-sequential ( set-graceful) graphs. Hence, while obtaining in general a ‘good’ characterization of a set-sequential (set-graceful) graphs appears a formidable open prob-lem [15] it becomes imperative to recognize graphs which are set-sequential (set-graceful). In particular, the problem of characterizing set-sequential trees was raised in [5] and [8] and was studied in [16] for the case of binary trees. The first result in the next section gives a new necessary condition for a graph to be set-sequential followed by a complete characterization of set-sequential caterpillar of diameter four. We also identify and provide a complete characterization of a class of set-sequential tree of diameter four.

2 New results

Theorem 2.1. LetG be a set-sequential graph with respect to a setX. Then, for every edge ab∈E(G) with d(a) =d(b) one has d(a) +d(b)<2n−1₋₁ _unless _G_∼₌_K

5.

Proof. Suppose the theorem is false. Then, there exists a (p, q)-graph G having a set-sequential labeling f with respect to a set X having an edge ab with d(a) = d(b), yet

d(a) +d(b)≥2n−1₋₁_.

Firstly, it is not difficult to verify that no graph with at most four vertices satisfies all the given properties, viz., a set-sequential graph H having an edge ab with d(a) = d(b) and

d(a) +d(b) ≥ 2n−1 ₋₁_, _where _n _{is the cardinality of the ground set} _X _{with respect to} which there is a set-sequential labeling ofH.Therefore,Ghas at least five vertices whence, by invoking the Acharya-Hegde necessary condition for a graph to be set-sequential, we see thatn≥3.If n= 3, it may be verified that q ≤2. Further, no graph on at least five vertices and at most two edges is set-sequential. Therefore, n≥4.If n >4 then it is not difficult to verify that p would never be an integer solution to the quadratic inequality

p2+p+ 2−2n+1 ≥0

which is set up by the facts thatp+q= 2n₋_{1 and} _q _≤ p(p−₁₎

2 . Therefore,n= 4 whence we get p = 5.Since G is set-sequential, we have p+q = 2n₋_{1 which, when} _p _{= 5 and}

n= 4,yields thatq= 10 and hence G∼=K5 and we know that it is set-sequential [6].

A caterpillar is a tree with the property that the removal of its endpoints leaves a path, we characterize set-sequential caterpillars of diameter four. Set-sequential caterpillars of diameter five and six has been characterized in [10]. But before we proceed any further we make the following conventions: byT[a, b, c] we would mean a caterpillar whose underlying path is P = [u, v, w] and u, v, w have degrees a, b, c in T[a, b, c]. For example T[2,2,2] is a path of length four or T[3,3,3] is as shown in the Fig. 1. Further we denote the pendant vertices adjacent to u, v,and w respectively asui where 1≤i≤a−1, vj where

1≤j≤b−2 andwk where 1≤k≤c−1.

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u v w

Figure 1: T[3,3,3].

Proposition 2.1. If T[a, b, c] is set-sequential thena, b, c are all odd >1.

Proof. The proof follows from Theorem 1.3.

As the Proposition 2.1 suggests all the internal vertices of set-sequential caterpillar of diameter four are odd, we would require the following terminology and notions which would smoothen our presentation.

Let So(n,3) denote the set of all the 3-decompositions of an integern into odd parts

>1.Where decomposition [11] is an order dependent partition of an integern.For example

Example 2.1.

So(17,3) ={(11,3,3),(3,11,3),(3,3,11),(9,5,3),(9,3,5),(5,9,3),(5,3,9),

(3,9,5),(3,5,9),(7,7,3),(7,3,7),(3,7,7),(7,5,5),(5,7,5),(5,5,7)}.

We say that So(n,3) generates So(m,3), m > n if each (x1, x2, x3)∈So(m,3) can be

expressed as

(x₁, x₂, x₃) = (y₁, y₂, y₃) + (2r,0,0), or

= (y1, y2, y3) + (0,2r,0), or = (y1, y2, y3) + (0,0,2r), or = (y1, y2, y3) + (2r−2,2r−2,0), or = (y1, y2, y3) + (2r−2,0,2r−2), or = (y₁, y₂, y₃) + (0,2r−2_,₂r−2₎_{, or} = (y1, y2, y3) + (2r−3,2r−3,2r−2), or = (y1, y2, y3) + (2r−3,2r−2,2r−3), or = (y1, y2, y3) + (2r−2,2r−3,2r−3)

for somer >0 and some (y1, y2, y3)∈So(n,3).

Proposition 2.2. So(2n+3+ 1,3) is generated bySo(2n+2+ 1,3)for r=n+ 2, n≥1.

Proof. Let n ≥ 1 and let (x1, x2, x3) ∈ So(2n+3 + 1,3) be a partition of 2n+3 + 1

into three odd parts, each part being at least 3. Without loss of generality assume that

x1 ≥x2≥x3.Note that the difference between 2n+3+ 1 and 2n+2+ 1 is 2n+2.Our objec-tive is to show that there exists some (y1, y2, y3)∈So(2n+2+ 1,3) such that the difference

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Case 1: Suppose x1 ≥2n+2+ 3: Then we sety1 =x1−2n+2, y2 =x2, y3 =x3. The condition onx₁implies thaty₁ ≥3, and so we have the 3-tuple (y₁, y₂, y₃)∈So(2n+2+1,3).

Case 2: Supposex1 ≤2n+2+ 1: Consider two subcases, depending on the value ofx2: Subcase a: Suppose x2 ≥ 2n+1 + 3. Then, x1 ≥ x2 ≥ 2n+1 + 3. So, we can set (y₁, y₂, y₃) to be (x₁−2n+1_{, x}

2−2n+1, x3), which is a partition of 2n+2+ 1 into three odd parts, each being of size at least 3.

Subcase b: Suppose x2 ≤2n+1+ 1.Then, sincex1 ≤2n+2+ 1 and x2 ≤2n+1+ 1, we have thatx3 ≥2n+3+ 1−(2n+2+ 1 + 2n+1+ 1) = 2n+3−2n+2−2n+1−1≥2n+ 3.Hence,

x2 ≥2n+ 3. Also,x2, x3 ≤2n+1+ 1 imply that x1 ≥2n+3+ 1−(2n+1+ 1 + 2n+1+ 1)≥ 2n+2₋₁_≥₂n+1_{+ 3}_._{Hence, we can subtract from (}_x

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Theorem 2.2. T[a, b, c] is set-sequential if and only if (a, b, c) ∈ So(2n+2 + 1,3) for

some n≥1.

Proof. LetT[a, b, c] be set-sequential, then in view of Acharya-Hegde necessary condi-tions, the fact that there is no set-sequential tree of diameter four with respect to a set of cardinality three and Proposition 2.1, it is seen that (a, b, c)∈So(2n+2+ 1,3) for some

n≥1.

Conversely, letT[a, b, c] be such that (a, b, c)∈So(2n+2+ 1,3) for somen≥1.We shall

show thatT[a, b, c] is set-sequential. Letu, v, wbe the internal vertices ofT[a, b, c] and the pendant vertices adjacent tou, v,andwrespectively be denoted asui where 1≤i≤a−1,

vj where 1≤j≤b−2 andwk where 1≤k≤c−1.Since,d(u) +d(v) +d(w) =a+b+c=

2n+2_{+ 1 it is easily seen that} _T_[_{a, b, c}_{] is of the order 2}n+2_{, and the number of pendant} vertices are 2n+2₋₃_.

For n= 1, So(9,3) ={(3,3,3)}. ThusT0[3,3,3] is unique caterpillar of diameter four with 21+2−3 = 5 pendant vertices and is seen to be set-sequential with respect to with respect toX1={x1, x2, x3} ∪ {x4}of cardinality 3 + 1 = 4, as shown in Table 1.

f1:V(T0[3,3,3])→ P(X1)− {∅}

V(T0[3,3,3]) f1(V(T0[3,3,3])

u, v, w f1(u) ={x1}, f1(v) ={x2}, f1(w) ={x3}

ui,1≤i≤2 f1(u1) ={x2, x3, x4}, f1(u2) ={x1, x2, x4}

vj, j= 1 f1(v1) ={x1, x2, x3}

wk,1≤k≤2 f1(w1) ={x4} ,f1(w2) ={x1, x3, x4}

Table 1: Set-sequential labeling ofT0[3,3,3].

For n = 2, So(17,3) is shown in example 2.1 and for (a, b, c) ∈ So(17,3), the non

isomorphic caterpillars of diameter four shown in the Table 2 and each of them contains 22+2−3 = 13 pendant vertices.

(a, b, c)∈So(17,3) All non isomorphic caterpillarsT[a, b, c] such that (a, b, c)∈So(17,3).

(3,3,11) or (11,3,3) T1[3,3,11], (3,11,3) T2[3,11,3], (9,5,3) or (3,5,9) T3[3,5,9], (5,9,3) or (3,9,5) T4[5,9,3], (9,3,5) or (5,3,9) T5[9,3,5], (3,7,7) or (7,7,3) T6[3,7,7], (7,3,7) T7[7,3,7], (5,5,7) or (7,5,5) T8[5,5,7], (5,7,5) T9[5,7,5].

Table 2: Non isomorphic caterpillarsT[a, b, c] associated with (a, b, c)∈So(17,3).

And, each ofTi[a, b, c],1≤i≤9 such that (a, b, c)∈So(17,3) seen to be set-sequential

with respect to X₂ =X₁∪ {x₅} ={x₁, x₂, x₃} ∪ {x₄} ∪ {x₅} of cardinality 3 + 2 = 5,as shown in the Table 3 - 4. Hence the result is true for n= 2.

Note that, each of Ti[a, b, c], 1 ≤ i ≤ 9 such that (a, b, c) ∈ So(17,3) has exactly

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f2:V(T1[3,3,11])→ P(X2)− {∅}

V(T1[3,3,11]) f2(V(T1[3,3,11]))

u, v, w f2(u) =f1(u), f2(v) =f1(v), f2(w) =f1(w)

ui, 1≤i≤2 f2(u1) =f1(u1), f2(u2) =f1(u2)

vj, j = 1 f2(v1) =f1(v1)

wk,1≤k≤10 f2(w1) =f1(w1) ,f2(w2) =f1(w2), f2(w3) ={x5},f2(w4) ={x1, x5},

f2(w5) ={x2, x5} ,f2(w6) ={x4, x5}, f2(w7) ={x1, x2, x5} ,

f2(w8) ={x1, x4, x5}, f2(w9) ={x2, x4, x5} ,f2(w10) ={x1, x2, x4, x5}.

f2:V(T2[3,11,3])→ P(X2)− {∅}

V(T2[3,11,3]) f2(V(T2[3,11,3]))

ui, 1≤i≤2 f2(u1) =f1(u1), f2(u2) =f1(u2)

vj, 1≤j≤9 f2(v1) =f1(v1), f2(v2) ={x5}, f2(v3) ={x1, x5},f2(v4) ={x3, x5},

f2(v5) ={x4, x5},f2(v6) ={x1, x3, x5}, f2(v7) ={x1, x4, x5},

f2(v8) ={x3, x4, x5}, f2(v9) ={x1, x3, x4, x5}.

wk,1≤k≤2 f2(w1) =f1(w1) ,f2(w2) =f1(w2)

f2:V(T3[3,5,9])→ P(X2)− {∅}

V(T3[3,5,9]) f2(V(T3[3,5,9]))

ui, 1≤i≤2 f2(u1) =f1(u1), f2(u2) =f1(u2)

vj, 1≤j≤3 f2(v1) =f1(v1), f2(v2) ={x5} ,f2(v3) ={x3, x5},

wk,1≤k≤8 f2(w1) =f1(w1) ,f2(w2) =f1(w2), f2(w3) ={x1, x5} ,f2(w4) ={x4, x5},

f2(w5) ={x1, x2, x5}, f2(w6) ={x1, x4, x5}, f2(w7) ={x2, x4, x5},

f2(w8) ={x1, x2, x4, x5}.

f2:V(T4[5,9,3])→ P(X2)− {∅}

V(T4[5,9,3]) f2(V(T4[5,9,3]))

ui, 1≤i≤4 f2(u1) =f1(u1), f2(u2) =f1(u2), f2(u3) ={x5},f2(u4) ={x2, x5}

vj, 1≤j≤7 f2(v1) =f1(v1), f2(v2) ={x3, x5}, f2(v3) ={x4, x5}, f2(v4) ={x1, x3, x5},

f2(v5) ={x1, x4, x5}, f2(v6) ={x3, x4, x5}, f2(v7) ={x1, x3, x4, x5}.

wk,1≤k≤2 f2(w1) =f1(w1) ,f2(w2) =f1(w2)

f2:V(T5[9,3,5])→ P(X2)− {∅}

V(T5[9,3,5]) f2(V(T5[9,3,5]))

ui, 1≤i≤8 f2(u1) =f1(u1), f2(u2) =f1(u2), f2(u3) ={x2, x5},f2(u4) ={x4, x5}

f2(u5) ={x2, x3, x5},f2(u6) ={x2, x4, x5}, f2(u7) ={x3, x4, x5},

f2(u8) ={x2, x3, x4, x5}

vj, j = 1 f2(v1) =f1(v1)

wk,1≤k≤4 f2(w1) =f1(w1) ,f2(w2) =f1(w2), f2(w3) ={x5},f2(w4) ={x1, x5}

f2:V(T6[3,7,7])→ P(X2)− {∅}

V(T6[3,7,7]) f2(V(T6[3,7,7]))

ui, 1≤i≤2 f2(u1) =f1(u1), f2(u2) =f1(u2)

vj, 1≤j≤5 f2(v1) =f1(v1), f2(v2) ={x5}, f2(v3) ={x3, x5},f2(v4) ={x1, x4, x5}

f2(v5) ={x1, x3, x4, x5}.

wk,1≤k≤6 f2(w1) =f1(w1) ,f2(w2) =f1(w2), f2(w3) ={x1, x5}, f2(w4) ={x4, x5}

f2(w5) ={x1, x2, x5}, f2(w6) ={x2, x4, x5}.

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f2:V(T7[7,3,7])→ P(X2)− {∅}

V(T7[7,3,7]) f2(V(T7[7,3,7]))

ui,1≤i≤6 f2(u1) =f1(u1), f2(u2) =f1(u2), f2(u3) ={x5},f2(u4) ={x3, x5}

f2(u5) ={x2, x4, x5},f2(u6) ={x2, x3, x4, x5}

vj,1≤j≤1 f2(v1) =f1(v1)

wk,1≤k≤6 f2(w1) =f1(w1) ,f2(w2) =f1(w2), f2(w3) ={x2, x5} ,f2(w4) ={x4, x5}

f2(w5) ={x1, x2, x5},f2(w6) ={x1, x4, x5}.

f2:V(T8[5,5,7])→ P(X2)− {∅}

V(T8[5,5,7]) f2(V(T8[5,5,7]))

vj,1≤j≤3 f2(v1) =f1(v1), f2(v2) ={x4, x5},f2(v3) ={x3, x4, x5}

wk,1≤k≤6 f2(w1) =f1(w1) ,f2(w2) =f1(w2), f2(w3) ={x2, x5} ,f2(w4) ={x1, x2, x5}

f2(w5) ={x1, x4, x5}, f2(w6) ={x1, x2, x4, x5}.

f2:V(T9[5,7,5])→ P(X2)− {∅}

V(T9[5,7,5]) f2(V(T9[5,7,5]))

vj,1≤j≤5 f2(v1) =f1(v1), f2(v2) ={x4, x5}, f2(v3) ={x1, x4, x5}, f2(v4) ={x3, x4, x5},

f2(v5) ={x1, x3, x4, x5}

wk,1≤k≤4 f2(w1) =f1(w1), f2(w2) =f1(w2), f2(w3) ={x2, x5} ,f2(w4) ={x1, x2, x5}.

Table 4: Set-sequential labeling ofT7[7,3,7], T8[5,5,7], T9[5,7,5].

any Ti[a, b, c], 1≤i≤9 such that (a, b, c) ∈So(17,3) can be obtained from T0[3,3,3] by introducing 8 new pendant vertices and making them adjacent to the internal vertices of

T0[3,3,3] in a manner such that the degree of each internal vertex after such an addition is odd.

Let the result be true forn >1,that is, for (a, b, c)∈So(2n+2+ 1,3) all the caterpillars

T[a, b, c] of diameter four are set-sequential with respect to the setXn=Xn−1∪ {xn+3}=

{x₁, x₂, . . . , xn+3},and |Xn|=n+ 3.Clearly T[a, b, c] such that (a, b, c)∈So(2n+2+ 1,3)

is of the order 2n+2 _{and contains 2}n+2₋_{3 pendant vertices.}

We shall now prove that the result is true for n+ 1 > 1. That is, if (a′_{, b}′_{, c}′₎ _∈

So(2n+3 + 1,3) then, T[a′, b′, c′] is set-sequential with respect to Xn+1 = Xn∪ {xn+4}, where|Xn+1|=n+ 4.

Let T[a′_{, b}′_{, c}′_{] any caterpillar such that (}_a′_{, b}′_{, c}′₎_∈_S

o(2n+3+ 1,3) and Xn+1 =Xn∪

{xn+4},where |Xn+1|=n+ 4 be a non empty set. Clearly T[a′, b′, c′] has exactly 2n+3− 3−(2n+2₋_{3) = 2}n+2 _{pendant vertices more than} _T_[_{a, b, c}_]_. _Therefore _T_[_a′

, b′

, c′

] can be obtained from T[a, b, c] by introducing 2n+2 _{isolated vertices and making them adjacent} to any of the internal vertices u, v and w of T[a, b, c], which is set-sequential under the hypothesis, in any 3-even partition of 2n+2_,_{but in view of Proposition 2.2 it is enough to} consider the partition of 2n+2 _{of the form}_{₂n+2_,₀_,₀_}_or_{₂n+1_,₂n+1_,₀_}_or_{₂n_,₂n_,₂n+1_}_.

Case 1. When T[a′_{, b}′_{, c}′

] is obtained making 2n+2 _{isolated vertices adjacent to any} one of the internal vertices u, v or w of set-sequentially labeledT[a, b, c].Without loss of generality let it be adjacent to u, wherefn(u) =fn−1(u) =· · · =f1(u) ={x1}. T[a, b, c] being set-sequential with respect toXn, to show thatT[a′, b′, c′] which is obtained making

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with respect toXn+1=Xn∪ {xn+4},it is enough to show that the unlabeled 2n+2vertices adjacent tou can be assigned the subsets of P(Xn+1)− P(Xn) in an injective manner so

that edges adjacent to them receives distinct elements of P(Xn+1)− P(Xn) as the

sym-metric difference of its end vertices. Let Y = P(Xn+1)− P(Xn), then |Y| = 2n+3 and

there are exactly 2n+2_{subsets of}_X

n+1inY which contains{x1}and exactly 2n+2 subsets of Xn+1 inY which does not contains {x1}. Thus Y can be partitioned into two classes

A and B such that A contains all the subsets of Xn+1 in Y which contains {x1} and B contains all the subsets ofXn+1 inY which do not contains{x1},hence|A|=|B|= 2n+2. By assigning all the 2n+2 _{elements of} _B _{to the 2}n+2 _{unlabeled pendant vertices adjacent} to u in T[a′

, b′

, c′

] in one-to-one manner, T[a′

, b′

, c′

] is seen to be set-sequential. Similar argument follows for the case when T[a′_{, b}′_{, c}′

] is obtained from set-sequentially labeled

T[a, b, c] by making 2n+2 _{isolated vertices adjacent to either} _v _or _w _in _T_[_{a, b, c}_]_. _{Thus in} each of the caseT[a′

, b′

, c′

] is seen to be set sequential.

Case 2. When T[a′

, b′

, c′

] is obtained making 2n+1 _{isolated vertices adjacent to} any two of the internal vertices u, v, and w of set-sequentially labeled T[a, b, c]. With-out loss of generality let T[a′_{, b}′_{, c}′

] is obtained by making 2n+1 _{isolated vertices} ad-jacent to u and v of T[a, b, c], where fn(u) = fn−₁(u) = · · · = f₁(u) = {x₁} and

fn(v) = fn−1(v) = · · · = f1(v) = {x2}. Note these 2n+1 new pendant vertices can be partitioned into 2n−1 _{copies each of which contains four vertices. Let} _C

i and Ci′ denote

the partition of 2n+1 _{new pendant vertices adjacent to} _u _and _v _{each containing four} ver-tices, where 1≤i≤2n−₁

.To show that the unlabeled 2n+1 _{new pendant vertices adjacent} touandvcan be assigned the subsets ofP(Xn+1)− P(Xn) in an injective manner so that

edges adjacent to them receives distinct elements of P(Xn+1)− P(Xn) as the symmetric

difference of its end vertices, considerYn+1 =Xn+1− {x1, x2, x3, x4}={x5, x6, . . . , xn+4}, and Yn = Xn− {x1, x2, x3, x4} = {x5, x6, . . . , xn+3}, thus |Yn+1| = n and |Yn| = n−1

which implies |P(Yn+1)| = 2n and |P(Yn)| = 2n−1. Let Y′ = P(Yn+1) − P(Yn), thus

|Y′

| = |P(Yn+1)− P(Yn)| = 2n−1. Let Ai ∈ Y′; 1 ≤ i ≤ 2n−1. For each partition Ci of

vertices adjacent to u we make the following assignment:

Ci:

      

fn+1(u1,i) ={Ai}

fn+1(u2,i) ={x2} ∪ {Ai}

fn+1(u3,i) ={x3, x4} ∪ {Ai}

fn+1(u4,i) ={x2, x3, x4} ∪ {Ai}

and for each partition C′

i of vertices adjacent to v we make the following assignment:

C′ i :       

fn+1(v1,i) ={x3} ∪ {Ai}

fn+1(v2,i) ={x4} ∪ {Ai}

fn+1(v3,i) ={x1, x3} ∪ {Ai}

fn+1(v4,i) ={x1, x4} ∪ {Ai}

whereAi∈Y′ and 1≤i≤2n−1.

Clearly the assignment defined above is injective and is a set-sequential labeling ofT[a′_{, b}′_{, c}′_] with respect to Xn+1,when T[a′, b′, c′] is obtained by making 2n+1 isolated vertices ad-jacent to u and v of set-sequentially labeled T[a, b, c]. For the case when T[a′

, b′

, c′ ] is obtained by making 2n+1 _{isolated vertices adjacent to the internal vertices} _u _and _w _of set-sequentially labeled T[a, b, c], in the assignment described as above replacing x1 by

x3, x2 by x1 and x3 by x2 we get the set-sequential labeling of T[a′, b′, c′].Finally, when

T[a′

, b′

, c′

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x1 by x2, x2 by x3 and x3 by x1 we get the set-sequential labeling of T[a′, b′, c′].Thus in either of the casesT[a′_{, b}′_{, c}′

] is seen to be set-sequential.

Case 3. WhenT[a′_{, b}′_{, c}′_{] is obtained making 2}n _{isolated vertices adjacent to any two}

of the internal vertices u, v, and w and 2n+1 _{isolated vertices adjacent to any one of the} remaining internal verticesu, v,andwof set-sequentially labeledT[a, b, c].Without loss of generality letT[a′_{, b}′_{, c}′_{] is obtained by making 2}n _{isolated vertices adjacent to}_v _and_w _of

T[a, b, c] and 2n+1 _{isolated vertices adjacent to} _u,_where_f

n(u) =fn−1(u) =· · ·=f1(u) =

{x1}, fn(v) = fn−1 = · · · = f1(v) = {x2} and fn(w) = fn−1(w) = · · · = f1(w) = {x3}. Since, 2n+1 _{new pendant vertices can be partitioned into 2}n−₁

copies each of which con-tains four vertices and 2n _{new pendant vertices can be partitioned into 2}n−1 _{copies each} of which contains two vertices. Let Ci denote the partition of unlabeled 2n+1 pendant

vertices adjacent to u each containing four vertices, where 1≤ i≤ 2n−₁

and Cj and Cj′

denote the partition of unlabeled 2n_{pendant vertices adjacent to}_v_and _w_{each containing}

two vertices, where 1≤j ≤2n−1_._{For each partition}_C

i of vertices adjacent to u we make

the following assignment:

Ci:



  

  

fn+1(u1,i) ={x3} ∪ {Ai}

fn+1(u2,i) ={x2, x3} ∪ {Ai}

fn+1(u3,i) ={x2, x4} ∪ {Ai}

fn+1(u4,i) ={x2, x3, x4} ∪ {Ai},

whereAi∈Y′ and 1≤i≤2n−1.For each partition Cj of vertices adjacent tov we make

Cj :

{

fn+1(v1,j) ={Aj}

fn+1(v2,j) ={x1} ∪ {Aj}

where Aj ∈Y′ and 1 ≤j≤2n−1, and for each partitionCj′ of vertices adjacent to w we

make the following assignment:

C′

j :

{

fn+1(w1,j) ={x4} ∪ {Aj}

fn+1(w2,j) ={x1, x4} ∪ {Aj}

whereAj ∈Y′ and 1≤j≤2n−1.

Clearly the assignment defined above is injective and is a set-sequential labeling ofT[a′

, b′

, c′ ] with respect toXn+1,whenT[a′, b′, c′] is obtained by making 2nisolated vertices adjacent to v and w and 2n+1 _{isolated vertices adjacent to} _u _{of set-sequentially labeled} _T_[_{a, b, c}_]_. For the case when T[a′

, b′

, c′

] is obtained by making 2n _{isolated vertices adjacent to the}

internal vertices u and w and 2n+1 _{isolated vertices adjacent to} _v _{of of set-sequentially} labeledT[a, b, c], in the assignment described as above replacingx₁ byx₂, x₂byx₃andx₃

byx1 we get the set-sequential labeling ofT[a′, b′, c′].Finally, whenT[a′, b′, c′] is obtained by making 2nisolated vertices adjacent to the internal verticesu andv, and 2n+1 isolated vertices adjacent to wof of set-sequentially labeled T[a, b, c], in the assignment described as above replacing x1 by x3, x2 by x1 and x3 by x2 we get the set-sequential labeling of

T[a′_{, b}′_{, c}′

].Thus in either of the cases T[a′_{, b}′_{, c}′

] is seen to be set-sequential.

Corollary 2.1. T[a, b, c] +K1 is set-graceful if and only if (a, b, c) ∈So(2n+2+ 1,3)

for some n≥1.

We now introduce a class of tree of diameter four. By a star lobster we would mean a tree with the property that the removal of its endpoints leaves a star K1,n,denoted as

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are not the end vertices or central vertex. It is easy to see that star lobster is of diameter 4. Set-sequential L[d₁, d₂, . . . , dn] fordi= 3 : 1≤i≤nis characterized as

Theorem 2.3. [8] L[d₁, d₂, . . . , dm]for di= 3; 1≤i≤m is set-sequential if and only if

3m≡0(mod 2n₋₁₎_.

Using arguments similar to proposition 2.2 it is verifiable that the following is true

Proposition 2.3. So(2n+4−1,3)is generated by So(2n+3−1,3)for n≥1.

Our next result characterizes set-sequential L[a, b, c].

Theorem 2.4. L[a, b, c] is set-sequential if and only if (a, b, c) ∈ So(2n+3 −1,3) for

some n≥1.

Proof. The condition is necessary in view of Acharya-Hegde necessary condition and Theorem 1.3.

Conversely, let L[a, b, c] be such that (a, b, c)∈So(2n+3−1,3) for some n≥1.Let the

central vertex ofL[a, b, c] be denoted as z and theu, v, w be the vertices of which are not the pendant vertices, and pendant vertices adjacent tou, v,andwrespectively be denoted as ui where 1 ≤i ≤a−1, vj where 1 ≤j ≤b−1 and wk where 1 ≤ k≤ c−1. Since,

d(u) +d(v) +d(w) +d(z) =a+b+c+ 3 = 2n+3_{+ 2 it is easily seen that}_L_[_{a, b, c}_{] is of the} order 2n+3_{, and the number of pendant vertices are 2}n+3₋₄_.

We shall show that L[a, b, c] is set-sequential.

For n = 1, all the non-isomorphic tress associated with So(15,3) are shown in the

Table 5 and each of them has 21+3₋_{4 = 12 pendant vertices.}

Non isomorphic trees L[a, b, c] such that (a, b, c)∈So(15,3)

L1

1[3,3,9], L 1 2[3,9,3]

L1

3[3,5,7], L 1

4[3,7,5], L 1 5[5,3,7]

L1 6[5,5,5]

Table 5: All Non isomorphic treesL[a, b, c] such that (a, b, c)∈So(15,3).

They are seen to be set-sequential with respect to X1={x1, x2, x3, x4} ∪ {x5},|X1|= 4 + 1 = 5 as shown in Table 6.

For n= 2,all the non-isomorphic tress associated with So(31,3) and are shown in the

Table 7 and each of them has 22+3−4 = 28 pendant vertices. Thus each of L2_i[a, b, c], 1≤i≤31 where (a, b, c)∈So(31,3) has exactly 22+3−4−(21+3−4) = 22+2= 16 pendant

vertices more than anyL1_j[a′_{, b}′_{, c}′

], 1≤j ≤6 where (a′_{, b}′_{, c}′

)∈So(15,3).

Note that except for L2₂₃[5,9,17], L2₂₄[9,17,5], L2₂₅[17,5,9] each of L2_i[a, b, c] such that (a, b, c)∈So(31,3) can be obtained fromL1j; 1≤j≤6 by making adjacent 16 new pendant

vertices to u, v, w in L1_j, by considering any partition of 21+3 = 16 of the form {16,0,0}

or {8,8,0} or {4,4,8}.Also the vertices u, v, w inL1_j[a, b, c] are labeled as f1(u) ={x1},

f1(v) = {x2} and f1(w) = {x3} which is same as the labels of the internal vertices of

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f1:V(L 1

1[3,3,9])→ P(X1)− {∅}

V(L1

1[3,3,9]) f1(V(L 1

1[3,3,9]))

z, u, v, w f1(z) ={x1, x2, x3}, f1(u) ={x1}, f1(v) ={x2}, f1(w) ={x3}

ui,1≤i≤2 f1(u1) ={x4}, f1(u2) ={x3, x4}

vj,1≤j≤2 f1(v1) ={x5}, f1(v2) ={x3, x5}

wk,1≤k≤8 f1(w1) ={x1, x5}, f1(w2) ={x4, x5}, f1(w3) ={x1, x2, x4}

f1(w4) ={x1, x2, x5}, f1(w5) ={x1, x4, x5}, f1(w6) ={x2, x3, x4}

f1(w7) ={x2, x4, x5}, f1(w8) ={x1, x2, x4, x5}.

f1:V(L12[3,9,3])→ P(X1)− {∅}

V(L1

2[3,9,3]) f1(V(L 1

2[3,9,3]))

ui,1≤i≤2 f1(u1) ={x4}, f1(u2) ={x3, x4}

vj,1≤j≤8 f1(v1) ={x5}, f1(v2) ={x1, x5}, f1(v3) ={x3, x5}, f1(v4) ={x4, x5},

f1(v5) ={x1, x3, x5}, f1(v6) ={x1, x4, x5}, f1(v7) ={x3, x4, x5},

f1(v8) ={x1, x3, x4, x5}.

wk,1≤k≤2 f1(w1) ={x1, x2, x4}, f1(w2) ={x2, x3, x4}

f1:V(L13[3,5,7])→ P(X1)− {∅}

V(L1

3[3,5,7]) f1(V(L 1

3[3,5,7]))

ui,1≤i≤2 f1(u1) ={x4}, f1(u2) ={x3, x4}

vj,1≤j≤4 f1(v1) ={x5}, f1(v2) ={x3, x5}, f1(v3) ={x1, x4, x5},

f1(v4) ={x1, x3, x4, x5},

wk,1≤k≤6 f1(w1) ={x1, x5}, f1(w2) ={x4, x5}, f1(w3) ={x1, x2, x4},

f1(w4) ={x1, x2, x5}, f1(w5) ={x2, x3, x4}, f1(w6) ={x2, x4, x5}.

f1:V(L14[3,7,5])→ P(X1)− {∅}

V(L1

4[3,7,5]) f1(V(L 1

4[3,7,5]))

ui,1≤i≤2 f1(u1) ={x4}, f1(u2) ={x3, x4}

vj,1≤j≤6 f1(v1) ={x1, x5}, f1(v2) ={x4, x5}, f1(v3) ={x1, x3, x5}

f1(v4) ={x1, x4, x5}, f1(v5) ={x3, x4, x5}, f1(v6) ={x1, x3, x4, x5}

wk,1≤k≤4 f1(w1) ={x5}, f1(w2) ={x2, x5}, f1(w3) ={x1, x2, x4},

f1(w4) ={x2, x3, x4}.

f1:V(L 1

5[5,3,7])→ P(X1)− {∅}

V(L1

5[5,3,7]) f1(V(L 1

5[5,3,7]))

ui,1≤i≤4 f1(u1) ={x4}, f1(u2) ={x5}, f1(u3) ={x3, x4}, f1(u4) ={x3, x5},

vj,1≤j≤2 f1(v1) ={x4, x5}, f1(v2) ={x3, x4, x5},

wk,1≤k≤6 f1(w1) ={x2, x5}, f1(w2) ={x1, x2, x4}, f1(w3) ={x1, x2, x5}

f1(w4) ={x1, x4, x5}, f1(w5) ={x2, x3, x4}, f1(w6) ={x1, x2, x4, x5}.

f1:V(L 1

6[5,5,5])→ P(X1)− {∅}

V(L1

6[5,5,5]) f1(V(L16[5,5,5]))

vj,1≤j≤4 f1(v1) ={x4, x5}, f1(v2) ={x1, x4, x5}, f1(v3) ={x3, x4, x5},

f1(v4) ={x1, x3, x4, x5},

f1(w4) ={x2, x3, x4}.

Table 6: Set-sequential labeling ofL1

1[3,3,9], L 1

2[3,9,3], L 1

3[3,5,7], L 1

4[3,7,5], L 1

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Non isomorphic trees L[a, b, c] such that (a, b, c)∈So(31,3)

L2

1[3,3,25], L 2

2[3,25,3]

L2

3[3,5,23], L 2

4[5,23,3], L 2

5[23,3,5]

L2

6[3,7,21], L 2

7[7,21,3], L 2

8[21,3,7]

L2

9[3,9,19], L 2

10[9,19,3], L 2

11[19,3,9]

L2

12[3,11,17], L 2

13[11,17,3], L 2

14[17,3,11]

L2

15[3,13,15], L 2

16[13,15,3], L 2

17[15,3,13]

L2

18[5,5,21], L 2

19[5,21,5]

L2

20[5,7,19], L 2

21[7,19,5], L 2

22[19,5,7]

L2

23[5,9,17], L 2

24[9,17,5], L 2

25[17,5,9]

L2

26[5,11,15], L 2

27[11,15,5], L 2

28[15,5,11]

L2

29[5,13,13], L 2

30[13,5,13]

L2

31[7,7,17], L 2

32[7,17,7]

L2

33[7,9,15], L 2

34[9,15,7], L 2

35[15,7,9]

L2

36[7,11,13], L 2

37[11,13,7], L 2

38[13,7,11]

L2

39[9,9,13], L 2

40[9,13,9]

L2

41[9,11,11], L 2

42[11,9,11]

Table 7: All the non isomorphic treesL[a, b, c] associated with (a, b, c)∈So(31,3).

Thus all the trees L[a, b, c] for (a, b, c)∈S0(31,3) are seen to be set-sequential for the set X2 =X1∪ {x6}={x1, x2, x3, x4} ∪ {x5} ∪ {x6},|X2|= 4 + 2 = 6.

Let the result be true for n > 1, that is, for (a, b, c) ∈ So(2n+3−1,3) all the star

lobsters L[a, b, c] are set-sequential with respect to the set Xn = Xn−1∪ {xn+4}, where

|Xn|= 4 +n=n+ 4.ClearlyL[a, b, c] is of the order 2n+3 and contains 2n+3−4 pendant

vertices.

We shall now prove that the result is true for n+ 1 > 1. That is, if (a1, b1, c1) ∈

So(2n+4−1,3) then, L[a1, b1, c1] is set-sequential with respect to Xn+1 =Xn∪ {xn+5}, where|Xn+1|= 4 + (n+ 1) =n+ 5.

Let L[a1, b1, c1] be such that (a1, b1, c1) ∈ So(2n+4 −1,3) and Xn+1 = Xn∪ {xn+5}, where|Xn+1|=n+ 5 be any set.

Clearly L[a₁, b₁, c₁] has exactly 2n+4 ₋₄₋₍₂n+3₋_{4) = 2}n+3 _{pendant vertices more} than any of L[a, b, c] such that (a, b, c) ∈ So(2n+3 −1,3). Therefore L[a1, b1, c1] can be obtained from L[a, b, c] by introducing 2n+3 _{isolated vertices and making them adjacent} to any of the verticesu, v andwof L[a, b, c],which is set-sequential under the hypothesis, in any 3-even partition of 2n+3_,_{but in view of Proposition 2.3 it is enough to consider the} partition of 2n+3 _{of the form} _{₂n+3_,₀_,₀_}_or_{₂n+2_,₂n+2_,₀_{} {}₂n+1_,₂n+1_,₂n+2_}_.

Case 1. When L[a1, b1, c1] is obtained by making 2n+3 isolated vertices adjacent to any one of the vertex u, v, w of set-sequentially labeled L[a, b, c].With argument similar as in case (1) of Theorem 2.2, L[a1, b1, c1] is seen to be set-sequential.

Case 2. When L[a1, b1, c1] is obtained by making 2n+2 isolated vertices adjacent to any two of the vertices u, v, w of set-sequentially labeled L[a, b, c]. Without loss of gen-erality let L[a1, b1, c1] is obtained by making 2n+2 isolated vertices adjacent to u and v of L[a, b, c], where fn(u) = fn−1(u) = · · · = f1(u) ={x1}, and fn(v) = fn−1(v) = · · · =

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f2:V(L 2

23[5,9,17])→ P(X2)− {∅}

V(L2

23[5,9,17]) f2(V(L 2

23[5,9,17]))

vj,1≤j≤8 f2(v1) ={x6}, f2(v2) ={x4, x5}, f2(v3) ={x3, x6}, f2(v4) ={x5, x6}

f2(v5) ={x1, x4, x5}, f2(v6) ={x3, x4, x5}, f2(v7) ={x3, x5, x6}

f2(v8={x1, x3, x4, x5},

wk,1≤k≤16 f2(w1) ={x1, x6}, f2(w2) ={x2, x5}, f2(w3) ={x4, x6},

f2(w10) ={x2, x4, x6}, f2(w11) ={x4, x5, x6}, f2(w12) ={x1, x2, x4, x6},

f2(w13) ={x1, x2, x5, x6}, f2(w14) ={x1, x4, x5, x6},

f2(w15) ={x2, x4, x5, x6}, f2(w16) ={x1, x2, x4, x5, x6}.

f1:V(L 2

24[9,17,5])→ P(X2)− {∅}

V(L2

24[9,17,5]) f2(V(L 2

24[9,17,5]))

f2(u5) ={x3, x4}, f2(u6) ={x3, x5}, f2(u7) ={x5, x6},

f2(u8) ={x2, x5, x6},

vj,1≤j≤16 f2(v1) ={x3, x6}, f2(v2) ={x4, x5}, f2(v3) ={x4, x6},

f2(v10) ={x4, x5, x6}, f2(v11) ={x1, x3, x4, x5}, f2(v12) ={x1, x3, x4, x6},

f2(v13) ={x1, x3, x5, x6}, f2(v14) ={x1, x4, x5, x6},

f2(v15) ={x3, x4, x5, x6}, f2(v16) ={x1, x3, x4, x5, x6},

f2(w4) ={x2, x3, x4}.

f1:V(L 2

25[17,5,9])→ P(X2)− {∅}

V(L2

25[17,5,9]) f2(V(L 2

25[17,5,9]))

ui,1≤i≤16 f2(u1) ={x4}, f2(u2) ={x1, x5}, f2(u3) ={x2, x6},

f2(u4) ={x3, x4}, f2(u5) ={x3, x5}, f2(u6) ={x4, x6},

f2(u7) ={x2, x3, x6}, f2(u8) ={x2, x4, x6}, f2(u9) ={x2, x5, x6},

f2(u10) ={x3, x4, x6}, f2(u11) ={x4, x5, x6}, f2(u12) ={x2, x3, x4, x6},

f2(u13) ={x2, x3, x5, x6}, f2(u14) ={x2, x4, x5, x6},

f2(u15) ={x3, x4, x5, x6}, f2(u16) ={x2, x3, x4, x5, x6},

vj,1≤j≤4 f2(v1) ={x4, x5}, f2(v2) ={x1, x4, x5}, f2(v3) ={x3, x4, x5},

f2(v4) ={x1, x3, x4, x5},

wk,1≤k≤8 f2(w1) ={x5}, f2(w2) ={x1, x6}, f2(w3) ={x2, x5}, f2(w4) ={x5, x6},

f2(w8) ={x2, x3, x4},

Table 8: Set-sequential labeling ofL2

23[5,9,17], L 2

24[9,17,5], L 2

25[17,5,9].

pendant vertices adjacent to u and v each containing four vertices, where 1 ≤ i ≤ 2n_.

To show that the unlabeled 2n+2 _{new pendant vertices adjacent to} _u _and _v _{can be} assigned the subsets of P(Xn+1) − P(Xn) in an injective manner so that edges

adja-cent to them receives distinct elements of P(Xn+1)− P(Xn) as the symmetric

differ-ence of its end vertices, consider Yn+1 = Xn+1 − {x1, x2, x3, x4} = {x5, x6, . . . , xn+5}, and Yn = Xn− {x1, x2, x3, x4} = {x5, x6, . . . , xn+4}, thus |Yn+1| = n+ 1 and |Yn| = n

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|Y1| = |P(Yn+1)− P(Yn)| = 2n. Let Ai ∈ Y1; 1 ≤i ≤ 2n. For each each partitionCi of

vertices adjacent to u we make the following assignment:

Ci:



  

  

fn+1(u1,i) ={Ai}

fn+1(u2,i) ={x2} ∪ {Ai}

fn+1(u3,i) ={x3, x4} ∪ {Ai}

fn+1(u4,i) ={x2, x3, x4} ∪ {Ai}

and for each partition C′

i of vertices adjacent to v we make the following assignment:

C′

i :



  

  

fn+1(v1,i) ={x3} ∪ {Ai}

fn+1(v2,i) ={x4} ∪ {Ai}

fn+1(v3,i) ={x1, x3} ∪ {Ai}

fn+1(v4,i) ={x1, x4} ∪ {Ai}

whereAi∈Y′ and 1≤i≤2n.

Clearly the assignment defined above is injective and is a set-sequential labeling ofL[a1, b1, c1] with respect to Xn+1, when L[a1, b1, c1] is obtained by making 2n+2 vertices adjacent to

u and v of set-sequentially labeled L[a, b, c]. For the case when L[a1, b1, c1] obtained by making 2n+2 isolated vertices adjacent to the internal verticesu andw of set-sequentially labeled L[a, b, c], in the assignment described as above replacing x₁ by x₃, x₂ by x₁ and

x3 by x2 we get the set-sequential labeling of L[a1, b1, c1]. Finally, when L[a1, b1, c1] is obtained by making 2n+2 isolated vertices adjacent to the internal verticesvandwof set-sequentially labeledL[a, b, c], in the assignment described asx₁ byx₂, x₂ byx₃ andx₃ by

x1 we get the set-sequential labeling ofL[a1, b1, c1].Thus in either of the casesL[a1, b1, c1] is seen to be set-sequential.

Case 3. When L[a1, b1, c1] is obtained making 2n+1 isolated vertices adjacent to any two of the vertices u, v, and w and 2n+2 _{isolated vertices adjacent to any one of} the remaining vertices u, v, and w of set-sequentially labeled L[a, b, c]. Without loss of generality let L[a1, b1, c1] obtained by making 2n+1 isolated vertices adjacent to v and

w and 2n+2 _{isolated vertices adjacent to} _u _{of set-sequentially labeled} _L_[_{a, b, c}_{], where}

fn(u) = fn−₁(u) = · · · = f₁(u) = {x₁}, fn(v) = fn−₁(v) = · · · = f₁(v) = {x₂} and

fn(w) = fn−1(w) =· · · = f1(w) = {x3}.Since, 2n+2 new pendant vertices can be parti-tioned into 2n _{copies each of which contains four vertices and 2}n+1 _{new vertices can be} partitioned into 2n _{copies each of which contains two vertices. Let} _C

i denote the

par-tition of unlabeled 2n+2 _{new vertices adjacent to} _u _{each containing four vertices, where} 1≤i≤2n_and _C

j andCj′ denote the partition of unlabeled 2n+1 new vertices adjacent to

v and w each containing two vertices where 1≤j ≤2n_._{For each partition} _C

i of vertices

adjacent tou we make the following assignment:

Ci:



  

  

fn+1(u1,i) ={x3} ∪ {Ai}

fn+1(u2,i) ={x2, x3} ∪ {Ai}

fn+1(u3,i) ={x2, x4} ∪ {Ai}

fn+1(u4,i) ={x2, x3, x4} ∪ {Ai},

where Ai ∈ Y′ and 1 ≤i ≤2n. For each partition Cj of vertices adjacent to v we make

Cj :

{

fn+1(v1,j) ={Aj}

fn+1(v2,j) ={x1} ∪ {Aj}

where Aj ∈ Y′ and 1 ≤ j ≤ 2n, and for each partition Cj′ of vertices adjacent to w we

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C′

j :

{

fn+1(w1,j) ={x4} ∪ {Aj}

fn+1(w2,j) ={x1, x4} ∪ {Aj}

whereAj ∈Y′ and 1≤j≤2n.

Clearly the assignment defined above is injective and is a set-sequential labeling ofL[a1, b1, c1] with respect toXn+1,whenL[a1, b1, c1] is obtained by making 2n+1 isolated vertices adja-cent tovandwand 2n+2_{isolated vertices adjacent to}_u_{of set-sequentially labeled}_L_[_{a, b, c}_]_. For the case whenL[a1, b1, c1] is obtained by making 2n+1 isolated vertices adjacent to the verticesuandwand 2n+2 _{isolated vertices adjacent to vertex}_v _{of set-sequentially labeled}

L[a, b, c], in the assignment described as above replacing x1 byx2, x2 by x3 and x3 byx1 we get the set-sequential labeling of L[a1, b1, c1].Finally, whenL[a1, b1, c1] is obtained by making 2n+1 _{isolated vertices adjacent to the vertices} _u _and _v _{and 2}n+2 _{isolated vertices} adjacent to w of set-sequentially labeled L[a, b, c], in the assignment described as above replacingx1byx3, x2 byx1 andx3byx2 we get the set-sequential labeling ofL[a1, b1, c1]. Thus in either of the cases L[a1, b1, c1] is be set-sequential. Hence the proof follows.

As an immediate consequence of the foregoing two theorems and Theorem 1.1 we have the following

Corollary 2.2. L[d1, d2, . . . , dn] +K1 for di = 3; 1 ≤i≤n is set-graceful if and only

if 3m≡0(mod2n₋₁₎_.

Corollary 2.3. L[a, b, c] +K1 is set-graceful if and only if (a, b, c) ∈ So(2n+3−1,3)

for some n≥1.

Note that in Case (1) of the proof of Theorem 2.2, if 2n+2 _{pendant vertices are made} adjacent to the end vertex w1 of the diametral path of set-sequentially labeled T[a, b, c] with respect to a set Xn =Xn−1∪ {xn}={x1, x2, . . . , xn},and |Xn|=n+ 3, such that

f(w1) = {x4}, then we get a set-sequential caterpillar T[a, b, c, d] of diameter five with respect to a set Xn+1 =Xn∪ {xn+1}={x1, x2, . . . , xn+1}, and |Xn|=n+ 4 and one of

the end vertex of a diametral path inT[a, b, c, d] is necessarily assigned with{xn+1}.Thus if T is a set-sequentially labeled caterpillar of diameter d, then by adding 2m _pendant

vertices to the end vertex of the diametral path in T, where 2m ₌ _|_V₍_T₎_| _{we get a}

set-sequential caterpillar of diameterd+ 1.Also set-sequential trees up to diameter 3 has been completely characterized [8]. Thus following is immediate.

Theorem 2.5. For every positive integerdthere is a set-sequential tree of diameter d.

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