Vol 7, No 8 (2016)

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Available online through

ISSN 2229 – 5046

CONNECTED TOTAL DOMINATING SETS AND ITS POLYNOMIAL OF K

n

×

P

r

B. STEPHEN JOHN*

Associate Professor, Department of Mathematics, Annai Velankanni College, Tholayavattam, Tamil Nadu, India.

T. ANISHA ROSE (ST)

Department of Mathematics,

Annai Velankanni College, Tholayavattam, Tamil Nadu, India.

(Received On: 06-08-16; Revised & Accepted On: 30-08-16)

ABSTRACT

I

n this paper, we are going to study the connected total domination polynomial of 𝐾𝐾𝑛𝑛×𝑃𝑃_𝑟𝑟. The connected total domination polynomial of a graph 𝐺𝐺 of order 𝑛𝑛 is defined

(

,

)

(

,

)

,

) (

∑

=

G i

i ct

ct

x

i

G

d

i

G

D

γ

where 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑖𝑖) is the number of connected total dominating sets of 𝐺𝐺 with size𝑖𝑖 and γct(𝐺𝐺) is the connected total domination number of 𝐺𝐺.

Keywords: Connected total dominating set, connected total domination number, connected total domination

polynomial.

1. INTRODUCTION

Let 𝐺𝐺 = (𝑉𝑉,𝐸𝐸) be a simple graph of order n. For any vertex 𝑣𝑣 ∈ 𝑉𝑉, the open neighborhood of 𝑣𝑣 is the set 𝑁𝑁(𝑣𝑣) = {𝑢𝑢 ∈ 𝑉𝑉/𝑢𝑢𝑣𝑣 ∈ 𝐸𝐸} and the closed neighborhood of v is the set 𝑁𝑁[𝑣𝑣] =𝑁𝑁(𝑣𝑣) ∪{𝑣𝑣}. For a set 𝑆𝑆 ⊆ 𝑉𝑉, the open neighborhood of S is N(S) =∪ N(v) and the closed neighborhood of S is 𝑁𝑁[𝑆𝑆] =𝑁𝑁(𝑆𝑆)∪ 𝑆𝑆. A set 𝑆𝑆 ⊆ 𝑉𝑉 is a connected total dominating set of G if every vertex 𝑣𝑣 ∈ 𝑉𝑉is adjacent to atleast one element of 𝑆𝑆 and the induced sub graph <𝑆𝑆> is connected. The connected total domination number γct(G) is called a γ-set.

The polynomial, Dct(G, x)

∑

=

) (

)

,

(

G i

i ct

ct

x

i

G

d

γ

is defined as connected total domination polynomial of 𝐺𝐺. where 𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺,𝑖𝑖) is the number of connected total dominating sets with size 𝑖𝑖.

2. CONNECTED TOTAL DOMINATION POLYNOMIALS

Definition: 2.1 A graph 𝐺𝐺 consists of a pair (V(G), E(G) ), where 𝑉𝑉(𝐺𝐺) is a non empty finite set whose elements are called points (or) vertices and 𝐸𝐸(𝐺𝐺) is a set of unordered pairs of distinct elements of 𝑉𝑉(𝐺𝐺). The elements of 𝐸𝐸(𝐺𝐺) are called lines or edges of the graph 𝐺𝐺.

Definition: 2.2 If 𝑒𝑒= {𝑢𝑢,𝑣𝑣} is an edge of a graph 𝐺𝐺, written 𝑒𝑒 = 𝑢𝑢𝑣𝑣, we say that 𝑒𝑒 joins the vertices 𝑢𝑢 and 𝑣𝑣. Also we say that 𝑢𝑢 and 𝑣𝑣 are adjacent vertices, 𝑢𝑢 and 𝑣𝑣 are incident with 𝑒𝑒. If two vertices or not joined, then we say that they are not-adjacent.

Definition: 2.3 The graph 𝐺𝐺is complete if every two distinct vertices of 𝐺𝐺are adjacent. A complete graph with 𝑛𝑛 vertices is denoted by 𝐾𝐾𝑛𝑛.

Definition: 2.4 A walk of a graph 𝐺𝐺 is an alternating sequence of points and lines 𝑣𝑣₀,𝑥𝑥₁,𝑣𝑣₁,𝑥𝑥₂,𝑣𝑣₂, … ,𝑣𝑣𝑛𝑛−₁,𝑥𝑥𝑛𝑛,𝑣𝑣𝑛𝑛 beginning and ending with points such that each line 𝑥𝑥_𝑖𝑖 is incident with 𝑣𝑣_𝑖𝑖−₁and 𝑣𝑣_𝑖𝑖, A walk is called a path if all its points are distinct.

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Definition: 2.5 A subset 𝑆𝑆 of vertices in a graph 𝐺𝐺 is said to be a dominating set, if every vertex 𝑣𝑣 ∈ 𝑉𝑉 − 𝑆𝑆 is adjacent to atleast one element of 𝑆𝑆. A dominating set of 𝐺𝐺 is said to be a total dominating set, if every vertex 𝑣𝑣 ∈ 𝑉𝑉 is adjacent to atleast one element of 𝑆𝑆.

Definition: 2.6 A total dominating set of G is said to be a connected total dominating set, if the induced sub graph

<𝑆𝑆> of 𝐺𝐺 is connected.

Theorem: 2.7 Let 𝐺𝐺 = 𝐾𝐾_𝑛𝑛×𝑃𝑃₂, then the total connected domination polynomial of G is, 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑥𝑥) =𝑛𝑛𝑥𝑥2_{[1 +}_𝑥𝑥_]2(𝑛𝑛−1)_{− 𝑛𝑛 �}₂₍��_{𝑛𝑛 −}₁₎_𝐶𝐶

1 −2� 𝑥𝑥2𝑛𝑛−1−(𝑛𝑛 −1)𝑥𝑥2𝑛𝑛+ 2𝑥𝑥𝑛𝑛[1 +𝑥𝑥]𝑛𝑛

Proof: 𝐺𝐺₁=𝐾𝐾_𝑛𝑛 be the complete graph with n vertices, 𝐺𝐺₂=𝑃𝑃₂, its product 𝐺𝐺= 𝐺𝐺₁× 𝐺𝐺₂ is given in figure 1.1.

𝐺𝐺 =𝐾𝐾𝑛𝑛 ×𝑃𝑃2

Figure 1.1

The vertices of 𝐺𝐺 are denoted by {𝑣𝑣₁₁,𝑣𝑣₁₂, … ,𝑣𝑣₁_𝑛𝑛,𝑣𝑣₂₁,𝑣𝑣₂₂, … ,𝑣𝑣₂_𝑛𝑛} Let, 𝑆𝑆_𝑖𝑖 = {𝑉𝑉_{𝑘𝑘𝑖𝑖}/ 𝑘𝑘= 1,2; 𝑖𝑖= 1,2, . . . ,𝑛𝑛} and

𝑇𝑇𝑗𝑗 = {𝑉𝑉𝑗𝑗𝑘𝑘/ 𝑗𝑗= 1,2; 𝑘𝑘= 1,2, . . . ,𝑛𝑛}

The total connected dominating set with cardinality 2 are, 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, 2) =�{𝑆𝑆𝑖𝑖}/ 𝑖𝑖= 1,2, . . . ,𝑛𝑛}

Therefore, 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺, 2) = 𝑛𝑛

The total connected dominating set with cardinality 3 are, 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, 3) =�𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗}/ 𝑖𝑖,𝑗𝑗= 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗}

Therefore, 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺, 3) = 𝑛𝑛 �2(��𝑛𝑛 −1)𝐶𝐶₁�

The total connected dominating set with cardinality 4 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, 4) =�𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘= 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘}

Therefore,𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺, 4) = 𝑛𝑛 �2(��𝑛𝑛 −1)𝐶𝐶₂�

The total connected dominating set with cardinality 5 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, 5) =�𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑙𝑙= 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘,𝑙𝑙}

Therefore,𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺, 5) = 𝑛𝑛 �2(��𝑛𝑛 −1)𝐶𝐶₃� Proceeding in this way, we get

The total connected dominating set with cardinality n are,

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Therefore,𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑛𝑛) = 𝑛𝑛 ��2(𝑛𝑛 −1)𝐶𝐶_n₋₂�+ 2

The total connected dominating set with cardinality 𝑛𝑛+ 1 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑛𝑛+ 1) = {𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘, … ,𝑥𝑥𝑐𝑐,𝑥𝑥𝑐𝑐+1}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 1 = 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 1 } ∪�𝑇𝑇_𝑗𝑗⋃{𝑥𝑥_𝑘𝑘}/ {𝑥𝑥_𝑘𝑘}∉ 𝑇𝑇_𝑗𝑗, 𝑗𝑗= 1,2; 𝑘𝑘= 1,2, … ,𝑛𝑛; 𝑗𝑗 ≠ 𝑘𝑘�

Therefore,𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑛𝑛+ 1) = 𝑛𝑛 ��2(𝑛𝑛 −1)𝐶𝐶_n₋₁�+ 2(𝑛𝑛𝐶𝐶₁) The total connected dominating set with cardinality 𝑛𝑛+ 2 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑛𝑛+ 2) = {𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘, … ,𝑥𝑥𝑐𝑐+2}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 2 = 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 2 } ⋃�𝑇𝑇_𝑗𝑗⋃{𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙}/ {𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙}∉ 𝑇𝑇_𝑗𝑗, 𝑗𝑗= 1,2; 𝑘𝑘,𝑙𝑙= 1,2, … ,𝑛𝑛; 𝑗𝑗 ≠ 𝑘𝑘,𝑙𝑙� Therefore,𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺,𝑛𝑛+ 2) = 𝑛𝑛 ��2(𝑛𝑛 −1)𝐶𝐶_n�+ 2(𝑛𝑛𝐶𝐶₂)

Proceeding in this way, we get

The total connected dominating set with cordinality 2𝑛𝑛 −2 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, 2𝑛𝑛 −2) = {𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘, … ,𝑥𝑥𝑐𝑐, … ,𝑥𝑥𝛼𝛼}/ 𝑖𝑖,𝑗𝑗, … ,𝛼𝛼= 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗, … ,𝛼𝛼}

⋃�𝑇𝑇𝑗𝑗⋃{𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙, … ,𝑥𝑥𝛼𝛼}/ {𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙, … ,𝑥𝑥𝛼𝛼}∉ 𝑇𝑇𝑗𝑗,𝑗𝑗= 1,2; 𝑘𝑘,𝑙𝑙, … ,𝛼𝛼= 1,2, … ,𝑛𝑛; 𝑗𝑗 ≠ 𝑘𝑘,𝑙𝑙, … ,𝛼𝛼�

Therefore,𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺, 2𝑛𝑛 −2) = 𝑛𝑛 ��2(𝑛𝑛 −1)𝐶𝐶₂₍_n₋₁₎₋₂�+ 2(𝑛𝑛𝐶𝐶_𝑛𝑛−₂) The total connected dominating set with cardinality 2𝑛𝑛 −1 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, 2𝑛𝑛 −1) = {𝑆𝑆𝑖𝑖−{𝑥𝑥𝑖𝑖}/{𝑥𝑥𝑖𝑖}∈ 𝑆𝑆𝑖𝑖; 𝑖𝑖= 1,2, . . . ,𝑛𝑛} Therefore, 𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺, 2𝑛𝑛 −1) = 2𝑛𝑛

The total connected dominating set with cardinality 2𝑛𝑛 are,

D

_ct

(

G

,

2 n

)

=

1

Therefore, 𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺, 2𝑛𝑛) = 1

Hence, the total connected domination polynomial of 𝐺𝐺 is,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑥𝑥) =𝑛𝑛𝑥𝑥2₊

_[

₂

₍

₁

₎

_]

1

C

n

−

𝑥𝑥3₊

_[

₂

₍

₁

₎

_]

2

C

n

−

𝑥𝑥4

… +�

n

[

2 (

n

−

1 )

C

₍_n₋₂₎

]

_{+ 2}_{� 𝑥𝑥}𝑛𝑛₊_�

n

[

2 (

n

−

1 )

C

₍_n₋₁₎

]

_{+ 2(}_{𝑛𝑛𝐶𝐶}₁₎_{� 𝑥𝑥}𝑛𝑛+1

… +�

n

[

2 (

n

−

1 )

C

₂₍_n₋₁₎₋₂

]

+ 2(𝑛𝑛𝐶𝐶𝑛𝑛−2)� 𝑥𝑥2𝑛𝑛−2+ 2𝑛𝑛𝑥𝑥2𝑛𝑛−1+𝑥𝑥2𝑛𝑛

⟹ 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑥𝑥) =� 𝑛𝑛𝑥𝑥

2₊

_[

₂

₍

₁

₎

_]

1

C

n

−

𝑥𝑥3₊

_[

₂

₍

₁

₎

_]

2

C

n

−

𝑥𝑥4₊_⋯₊

_[

₂

₍

₁

₎

_]

) 2 ( −

−

C

_n

n

𝑥𝑥𝑛𝑛

+

n

[

2 (

n

−

1 )

C

₍_n₋₁₎

]

𝑥𝑥𝑛𝑛+1₊_⋯₊

_[

₂

₍

₁

₎

_]

2 ) 1 ( 2 − −

−

C

_n

n

𝑥𝑥2𝑛𝑛−2_{+ 2}_{𝑛𝑛𝑥𝑥}2𝑛𝑛−1₊_𝑥𝑥2𝑛𝑛� +[2𝑥𝑥𝑛𝑛+ 2(𝑛𝑛𝐶𝐶₁)𝑥𝑥𝑛𝑛+1+⋯+ 2(𝑛𝑛𝐶𝐶_𝑛𝑛−₂)𝑥𝑥2𝑛𝑛−2]

⟹ 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑥𝑥) =�𝑛𝑛𝑥𝑥2₊

_[

₂

₍

₁

₎

_]

1

C

n

−

𝑥𝑥3₊

_[

₂

₍

₁

₎

_]

2

C

n

−

𝑥𝑥4₊_⋯₊

_[

₂

₍

₁

₎

_]

) 2 ( −

−

C

_n

n

𝑥𝑥𝑛𝑛

+

n

[

2 (

n

−

1 )

C

₍_n₋₁₎

]

𝑥𝑥𝑛𝑛+1₊_⋯₊

_[

₂

₍

₁

₎

_]

2 ) 1 ( 2 − −

−

C

_n

n

𝑥𝑥2𝑛𝑛−2₊

_[

₂

₍

₁

₎

_]

1 ) 1 ( 2 − −

−

C

_n

n

𝑥𝑥2𝑛𝑛−1

+

n

[

2 (

n

−

1 )

C

₂₍_n₋₁₎

]

_𝑥𝑥2𝑛𝑛₋

n

[

2 (

n

−

1 )

C

₁

−

2 ]

𝑥𝑥2𝑛𝑛−1₋₍_{𝑛𝑛 −}₁₎_𝑥𝑥2𝑛𝑛_�

+ [2𝑥𝑥𝑛𝑛_{+ 2(}_{𝑛𝑛𝐶𝐶}

1)𝑥𝑥𝑛𝑛+1+⋯+ 2(𝑛𝑛𝐶𝐶𝑛𝑛−2)𝑥𝑥2𝑛𝑛−2]

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Theorem: 2.8 Let 𝐺𝐺 = 𝐾𝐾_𝑛𝑛 × 𝑃𝑃_𝑟𝑟, then the total connected domination polynomial of 𝐺𝐺 is,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑥𝑥) =𝑛𝑛𝑥𝑥𝑟𝑟_{[1 +}_𝑥𝑥_]𝑟𝑟(𝑛𝑛−1)₋

_[

₍

₁

₎

_]

1

r

C

n

r

n

−

𝑥𝑥𝑟𝑟𝑛𝑛 −1₋₍_{𝑛𝑛 −}₁₎_𝑥𝑥𝑟𝑟𝑛𝑛 ₊

(𝑟𝑟 −2)[𝑛𝑛(𝑛𝑛 −1)]𝑥𝑥𝑟𝑟+1[1 +𝑥𝑥](𝑛𝑛−2)𝑟𝑟+𝑥𝑥(𝑟𝑟−2)𝑛𝑛[1 +𝑥𝑥]2𝑛𝑛 for some 𝑟𝑟 > 2.

Proof: 𝐾𝐾_𝑛𝑛 be the complete graph with 𝑛𝑛 vertices, 𝑃𝑃_𝑟𝑟is a path of length 𝑟𝑟. Then its product 𝐺𝐺 = 𝐾𝐾_𝑛𝑛 × 𝑃𝑃_𝑟𝑟is given in figure 1.2

𝐺𝐺 = 𝐾𝐾𝑛𝑛 × 𝑃𝑃𝑟𝑟 Figure: 1.2

The vertices of G is denoted by �𝑉𝑉_{𝑖𝑖𝑗𝑗}/ 𝑖𝑖= 1,2, … ,𝑟𝑟; 𝑗𝑗= 1,2, . . . ,𝑛𝑛} Let, 𝑆𝑆𝑖𝑖 = {𝑉𝑉𝑘𝑘𝑖𝑖/ 𝑘𝑘= 1,2, … ,𝑟𝑟; 𝑖𝑖= 1,2, . . . ,𝑛𝑛} and

𝑇𝑇𝑗𝑗 =�𝑉𝑉𝑗𝑗𝑘𝑘/𝑗𝑗= 1,2, … ,𝑟𝑟; 𝑘𝑘= 1,2, . . . ,𝑛𝑛}

The total connected dominating set with cardinality 𝑟𝑟 are, 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟) = {{𝑆𝑆𝑖𝑖}/ 𝑖𝑖= 1,2, . . . ,𝑛𝑛}

Therefore, 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟) = 𝑛𝑛

The total connected dominating set with cardinality 𝑟𝑟+ 1 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟+ 1) =�𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗}/ 𝑖𝑖,𝑗𝑗= 1,2, . . . ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗}⋃�𝑆𝑆𝑖𝑖−{𝑣𝑣𝑘𝑘𝑖𝑖}⋃{𝑣𝑣𝑘𝑘𝑗𝑗}/𝑘𝑘= 1,2, . . . ,𝑟𝑟 ; 𝑖𝑖,𝑗𝑗= 1,2, … ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗 }

Therefore, 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟+ 1) = 𝑛𝑛 �𝑟𝑟��(𝑛𝑛 −1)𝐶𝐶₁�+ (𝑟𝑟 −2)[𝑛𝑛(𝑛𝑛 −1)] The total connected dominating set with cardinality 𝑟𝑟+ 2 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟+ 2) =�𝑆𝑆𝑖𝑖⋃ {𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘= 1,2, . . . ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘 }⋃

�𝑆𝑆𝑖𝑖⋃𝑆𝑆𝑗𝑗− �𝑣𝑣𝑘𝑘𝑖𝑖,𝑣𝑣𝑘𝑘𝑗𝑗�⋃ {𝑥𝑥𝑖𝑖}/ 𝑘𝑘= 1,2, . . . ,𝑟𝑟 ; 𝑖𝑖,𝑗𝑗= 1,2, … ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗}

(5)

The total connected dominating set with cardinality (𝑟𝑟 −2)𝑛𝑛are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺, (𝑟𝑟 −2)𝑛𝑛) =�𝑆𝑆𝑖𝑖⋃ {𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘, … ,𝑥𝑥𝑐𝑐}/𝑖𝑖,𝑗𝑗,𝑘𝑘, … ,𝑐𝑐= 1,2, . . . ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘, … ,𝑐𝑐}

⋃�𝑆𝑆𝑖𝑖⋃𝑆𝑆_𝑗𝑗 − �𝑣𝑣𝑘𝑘𝑖𝑖,𝑣𝑣𝑘𝑘𝑗𝑗�⋃{𝑥𝑥𝑖𝑖, … ,𝑥𝑥𝑐𝑐}/ 𝑘𝑘= 1,2, . . . ,𝑟𝑟 ; 𝑖𝑖,𝑗𝑗, … ,𝑐𝑐= 1,2, … ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗, … ,𝑐𝑐} ⋃ {{𝑇𝑇_𝑗𝑗}/ 𝑗𝑗= 1,2, . . . ,𝑟𝑟 }

Therefore,𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺, (𝑟𝑟 −2)𝑛𝑛) =

n

[

r

(

n

−

1 )

C

₍_r₋₂₎₍_n₋_r₎

]

+ (𝑟𝑟 −2)[𝑛𝑛(𝑛𝑛 −1)]

[

(

n

−

2 )

rC

₍_r₋₂₎₍_n₋_r₋₁₎

]

+ 1

The total connected dominating set with cardinality (��𝑟𝑟 −2)𝑛𝑛+ 1 are,

𝐷𝐷𝑐𝑐𝑐𝑐�𝐺𝐺, (��𝑟𝑟 −2)𝑛𝑛+ 1�=�𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘, … ,𝑥𝑥𝑐𝑐,𝑥𝑥𝑐𝑐+1}/𝑖𝑖,𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 1 = 1,2, … ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 1} ⋃ �𝑆𝑆𝑖𝑖⋃𝑆𝑆_𝑗𝑗−{𝑣𝑣𝑘𝑘𝑖𝑖,𝑣𝑣𝑘𝑘𝑗𝑗}⋃{𝑥𝑥𝑖𝑖, … ,𝑥𝑥𝑐𝑐₊₁}/ 𝑘𝑘= 1,2, . . . ,𝑟𝑟 ; 𝑖𝑖,𝑗𝑗, … ,𝑐𝑐+ 1 = 1,2, … ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗, … ,𝑐𝑐+ 1} ⋃�𝑇𝑇𝑗𝑗⋃{𝑥𝑥𝑘𝑘}/ {𝑥𝑥𝑘𝑘} ∉ 𝑇𝑇𝑗𝑗; 𝑗𝑗= 1,2, . . . ,𝑟𝑟;𝑘𝑘= 1,2, … ,𝑛𝑛;𝑗𝑗 ≠ 𝑘𝑘�

Therefore,

𝑑𝑑𝑐𝑐𝑐𝑐�𝐺𝐺, (��𝑟𝑟 −2)𝑛𝑛+ 1�=

n

[

r

(

n

−

1 )

C

₍_r₋₂₎₍_n₊₁₋_r₎

]

+(𝑟𝑟 −2)[𝑛𝑛(𝑛𝑛 −1)]

(

n

−

2 )

rC

₍_r₋₂₎₍_n₋_r₎

]

_{+ (2}_𝑛𝑛)𝐶𝐶1

The total connected dominating set with cardinality (��𝑟𝑟 −2)𝑛𝑛+ 2 are,

𝐷𝐷𝑐𝑐𝑐𝑐�𝐺𝐺, (��𝑟𝑟 −2)𝑛𝑛+ 2�=�𝑆𝑆𝑖𝑖⋃{𝑥𝑥_𝑗𝑗, … ,𝑥𝑥𝑐𝑐₊₂}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 2 = 1,2, … ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘, … ,𝑐𝑐+ 2}⋃ �𝑆𝑆𝑖𝑖⋃𝑆𝑆𝑗𝑗−{𝑣𝑣𝑘𝑘𝑖𝑖,𝑣𝑣𝑘𝑘𝑗𝑗}⋃{𝑥𝑥𝑖𝑖, … ,𝑥𝑥𝑐𝑐+2}/ 𝑘𝑘= 1,2, . . . ,𝑟𝑟 ; 𝑖𝑖,𝑗𝑗, … ,𝑐𝑐+ 2 = 1,2, … ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗, … ,𝑐𝑐+ 2} ⋃�𝑇𝑇_𝑗𝑗⋃{𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙}/ {𝑥𝑥𝑘𝑘,𝑥𝑥𝑙𝑙} ∈ 𝑇𝑇_𝑗𝑗; 𝑗𝑗= 1,2, . . . ,𝑟𝑟;𝑘𝑘,𝑙𝑙= 1,2, … ,𝑛𝑛;𝑗𝑗 ≠ 𝑘𝑘,𝑙𝑙�

Therefore, 𝑑𝑑𝑐𝑐𝑐𝑐𝐺𝐺, (��𝑟𝑟 −2)𝑛𝑛+ 2) =

n

[

r

(

n

−

1 )

C

₍_r₋₂₎₍_n₊₂₋_r₎

]

+(𝑟𝑟 −2)[𝑛𝑛(𝑛𝑛 −1)]

(

n

−

2 )

rC

₍_r₋₂₎₍_n₋_r₊₁₎

]

+ (2𝑛𝑛)𝐶𝐶2 Proceeding in this way, we get

The total connected dominating set with cardinality𝑟𝑟𝑛𝑛 −2 are,

𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟𝑛𝑛 −2) =�𝑆𝑆𝑖𝑖⋃{𝑥𝑥𝑗𝑗,𝑥𝑥𝑘𝑘, … ,𝑥𝑥𝑐𝑐, … ,𝑥𝑥𝛼𝛼}/ 𝑖𝑖,𝑗𝑗,𝑘𝑘, … ,𝛼𝛼= 1,2, … ,𝑛𝑛 ; 𝑖𝑖 ≠ 𝑗𝑗,𝑘𝑘, … ,𝛼𝛼}⋃

�𝑆𝑆𝑖𝑖⋃𝑆𝑆_𝑗𝑗−{𝑣𝑣𝑘𝑘𝑖𝑖,𝑣𝑣𝑘𝑘𝑗𝑗}⋃{𝑥𝑥𝑖𝑖,𝑥𝑥_𝑗𝑗, … ,𝑥𝑥𝛼𝛼}/ 𝑘𝑘= 1,2, . . . ,𝑟𝑟 ; 𝑖𝑖,𝑗𝑗, … ,𝛼𝛼= 1,2, … ,𝑛𝑛; 𝑖𝑖 ≠ 𝑗𝑗, … ,𝛼𝛼}⋃ �𝑇𝑇_𝑗𝑗⋃{𝑥𝑥_𝑘𝑘,𝑥𝑥_𝑙𝑙, … ,𝑥𝑥_𝛼𝛼}/ {𝑥𝑥_𝑘𝑘,𝑥𝑥_𝑙𝑙, … ,𝑥𝑥_𝛼𝛼} ∉ 𝑇𝑇_𝑗𝑗; 𝑗𝑗= 1,2, . . . ,𝑟𝑟; 𝑘𝑘,𝑙𝑙= 1,2, … ,𝑛𝑛; 𝑗𝑗 ≠ 𝑘𝑘,𝑙𝑙, … ,𝛼𝛼� Therefore,

𝑑𝑑𝑐𝑐𝑐𝑐𝐺𝐺, (𝑟𝑟𝑛𝑛 −2) =

n r n

[ (

−

1)

C

r n( − −1) 2

]

+ (𝑟𝑟 −2)[𝑛𝑛(𝑛𝑛 −1)]

(

n

−

2 )

rC

(n−2)−(n−2)

]

+ (2n)C2n−2

The total connected dominating set with cardinality 𝑟𝑟𝑛𝑛 −1 are, 𝐷𝐷𝑐𝑐𝑐𝑐𝐺𝐺, (𝑟𝑟𝑛𝑛 −1) = {𝑆𝑆𝑖𝑖−{𝑥𝑥𝑖𝑖} / {𝑥𝑥𝑖𝑖}∈ 𝑆𝑆𝑖𝑖,𝑖𝑖= 1,2, … ,𝑛𝑛} Therefore, 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟𝑛𝑛 −1) = 𝑟𝑟𝑛𝑛

The total connected dominating set with cArdinality 𝑟𝑟𝑛𝑛 are, 𝐷𝐷𝑐𝑐𝑐𝑐(𝐺𝐺,𝑟𝑟𝑛𝑛) = 1

Therefore, 𝑑𝑑_{𝑐𝑐𝑐𝑐}(𝐺𝐺,𝑟𝑟𝑛𝑛) = 1

Hence the total connected domination polynomial of G is

[

]

r 1

1

]

(r

2)

[n(n

1)]

1)C

[r(n

n

)

,

(

G

x

=

nx

+

−

+

−

x

+

D

_ct r

+

[

−

+

−

]

+2

+

r 1 2

]

(r

2)[n(n

1)][(n

2)rC

1)C

[r(n

n

x

[

]

n r

x

( 2)

1) -r -2)(n -(r r)

-2)(n

-(r

]

(r

2)

[n(n

1)]

[

(n

2)rC

]

1 1)C

[r(n

n

....

+

−

+

−

+

−

[

]

1 2)n (r 1 r)

-2)(n -(r r)

-1 2)(n

-(r

]

(r

2)[n(n

1)][

(n

2)rC

]

2 1)C

[r(n

n

−

₊

+

−

+

+ +

+

nC

x

[

]

rn rn rn

n

x

rnx

x

nC

+

−

+

−

+

− −

−2 2 1

2 2)

-(n -2) -(n 2

-1)

-r(n

]

(r

2)

[n(n

1)]

[

(n

2)rC

]

2

(6)

[

1 2 ( 2)

1 2 (r-2)(n-r)

( , )

r

[r(n 1)C ]

r

n[r(n 1)C ]

r

.... n [r(n 1)C

]

r n

ct

D G x

nx

n

x

+

x

+

x

−

⇒

=

+

−

+

−

+ +

−

]

rn rn rn n r

x

rnx

x

n

x

+

−

+

−

+

− + − −

+ r(n-1)-2 2 1

1 ) 2 ( r) -1 2)(n

-(r

]

....

[r(n

1)C

]

1)C

[r(n

n

[

2 1 1

]

)

2 )][(

1 (

)[

2 (

)]

1 (

)[

2 (

−

+

−

+

r r

x

rC

n

r

x

n

r

n r r n r

x

rC

n

r

2 )[

(

1 )][(

2 )

₍ ₂₎₍ ₁₎

]

( 2)

(

....

+

−

₋ ₋ ₋ −

+

1 ) 2 ( ) )( 2 (

]

)

2 )][(

1 (

)[

2 (

−

₋ ₋ − +

+

r n

r n r

x

rC

n

r

( )1

_]

( 2) ( 2)

.... (

r

2)[ (

n n

1)][(

n

2)

rC

_n_{− − −}_n

]

x

rn− −r

+ + −

−

₊_� 2

2 2 1 ) 2 ( 1 ) 2 (

2 ...

2

− + ₋ −

−

₊

rn

n n r n r

x

nC

x

nC

x

�

[

r r r r n

ct

G

x

nx

n

x

D

(

,

)

=

+

[r(n

−

1)C

₁

]

+1

+

n

[r(n

−

1)C

₂

]

+2

+

....

+

n

[r(n

−

1)C

_(r_-_2)(n_-_r)

]

( −2)

⇒

1 1 -1) -r(n 2 2 -1) -r(n 1 ) 2 ( r) -1 2)(n

-(r

]

....

[r(n

1)C

]

[r(n

1)C

]

1)C

[r(n

n

−

₊ − +

+

−

+

−

+

r n rn rn

x

n

x

n

x

]

rn rn rn

x

n

x

n

x

n

[r(n

−

1)C

_r(n_-₁₎

]

−

[r(n

−

1)C

₁

-

r]

1

−

(

−

1 )

+

−

[

2 1 1

]

)

2 )][(

1 (

)[

2 (

)]

1 (

)[

2 (

−

+

−

+

r r

x

rC

n

r

x

n

r

n r r n r

x

rC

n

r

2 )[

(

1 )][(

2 )

₍ ₂₎₍ ₁₎

]

( 2)

(

....

+

−

₋ ₋₋ −

+

1 ) 2 ( ) )( 2 (

]

)

2 )][(

1 (

)[

2 (

−

₋ ₋ − +

+

r n

r n r

x

rC

n

r

( )1

_]

( 2) ( 2)

.... (

r

2)[ (

n n

1)][(

n

2)

rC

_n_{− − −}_n

]

x

rn− −r

+ + −

−

+� 2

2 2 1 ) 2 ( 1 ) 2 (

2 ...

2

− + ₋ −

−

₊

rn

n n r n r

x

nC

x

nC

x

�

Dct(G,x) =nxr [1+x]r(n-1)-

n

[

r

(

n

−

1 )

C

1−𝑟𝑟]x

rn-1

-(n-1)xrn + (r-2)[n(n-1)]xr+1[1+x](n-2)r + x(r-2)n [1+x]2n for some 𝑟𝑟> 2. Hence the Proof.

The following Table represents the coefficients of the total domination polynomial of 𝐺𝐺= 𝐾𝐾𝑛𝑛 × 𝑃𝑃₂ for all 𝑛𝑛 < 8. 𝑑𝑑𝑐𝑐𝑐𝑐(𝐺𝐺,𝑖𝑖)

𝐺𝐺 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16

𝐾𝐾3×𝑃𝑃2 𝐾𝐾4×𝑃𝑃2 𝐾𝐾5×𝑃𝑃2 𝐾𝐾6×𝑃𝑃2 𝐾𝐾7×𝑃𝑃2 𝐾𝐾8×𝑃𝑃2

3 4 5 6 7 8 14 24 40 60 84 112 24 62 140 270 462 728 6 88 282 720 1540 2912 1 72 360 1262 3465 8008 8 300 1524 5546 16016 1 160 1290 6482 24026 10 760 5586 27472 1 300 3535 24080 12 1610 16128 1 504 8148 14 3024 1

784 16 1

Table: 2.1

REFERENCES

1. Alikhani.S and Peng Y.H, Introduction to Domination Polynomial of a graph, arXiv: 0905-225[v] [math.co], 14 May (2009).

2. Alikhani.S, on the Domination Polynomial of some graph operations, ISRN combin, (2013).

3. Haynes .T.W, Hedtniemi. S.T and slater .P.J, Fundamentals of Domination in Graphs, Marcel Dekker, New York, (1998).

4. Saeid Alikhani and Emeric Deutsch, More on domination polynomial and domination root, arXiv:1305. 3734v2, (2014).

5. Sahib sh.kabat, Abdul Jalil M. Khalaf and Roslan Hasni, Dominating sets and Domination Polynomials of Wheels, Australian Journal of Applied Science, (2014).

6. Vijayan .A and Anitha Baby .T, Connected Total domination polynomials of graphs, International journal of mathematical Archieve, 5(11), (2014).

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8. Vijayan .A, Anitha Baby .T, Connected Total dominating sets and connected total domination polynomials of Gem graphs, IJSIMR journal of mathematics (2015).

Source of support: Nil, Conflict of interest: None Declared