Regular Number of Semitotal block graph of a graph

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International Journal of Research in Engineering & Applied Sciences

Email:- editorijrim@gmail.com, http://www.euroasiapub.org

70 Regular Number of Semitotal block graph of a graph

M.H.Muddebihal1_,

Professor,

Department of Mathematics,

Gulbarga University, Kalaburagi, 585 106 Karnataka, India

Abdul Gaffar2

Research Scholar,

Department of Mathematics, Gulbarga University,

Kalaburagi, 585 106 Karnataka, India

ABSTRACT

:

For any (p ,q) graph G, the vertices and blocks of a graph are called its members. The

semitotal block graph 𝑇𝑏 𝐺 of a graph G as the graph whose set of vertices is the union of the set of

vertices and blocks of G and in which two vertices are adjacent if and only if the corresponding

vertices of G are adjacent or the corresponding members are incident.. The regular number of a

𝑇𝑏(𝐺) is the minimum number of subsets into which the edge set of 𝑇𝑏(𝐺) should be partitioned so

that the subgraph induced by each subset is regular and is denoted by 𝑇𝑏(𝐺) . In this paper some

results on regular number of 𝑇𝑏(𝐺) were obtained and expressed in terms of elements of G.

1.Key words : Regular number / semitotal block graph /domination number / total domination

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International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR – 5.981)

71

2. Mathematics Subject Classification Number : 05C69, 05C70.

3. INTRODUCTION:

All graphs considered here are simple, finite, and non-trivial. As usual p and q denote the number

of vertices and edges of a graph G and the maximum degree of a vertex in G is denoted by ∆ G . A

graph G is called trivial if it has no edges. The maximum distance between any two vertices in a G is

called a diameter and is denoted by diam(G). The path and tree numbers were introduced by

Stanton James and Cown in10_{. Any undefined term in this paper may be found in}3_{. The edge set}

independence number 𝛽₁∗(G) is the minimum order of partition of E(G) into subsets so that the

subgraph induced by each set must be independent. The independence number 𝛽₁(𝐺) is the

maximum cardinality of an edge independent set in G. A set 𝐷′⊆ V is said to be a dominating set of

G, if every vertex in ( V – 𝐷′ ) is adjacent to some vertex in 𝐷′. The minimum cardinality of vertices

in such a set is called the domination number of G and is denoted by γ G . A dominating set is said

to be total dominating set of G, if N(𝐷′) = V or equivalently, if for every 𝑣∈𝑉, there exists a vertex u

∈ 𝐷′ , u ≠𝑣, such that u is adjacent to 𝑣. The total domination number of G, denoted by 𝛾_𝑡(𝐺) is the

minimum cardinality of total dominating set of G. Domination related parameters are now well

studied in graph theory. A set with minimum cardinality among all the maximal independent set of

G is called minimum independent dominating set of G. The cardinality of a minimum independent

dominating set is called independent domination number of the graph G and it is denoted by i(G).

Total domination in graphs was studied by E.J.Cockayne, R.M.Dawes, and S.T.Hedetniemi in1_{. This}

concept was studied by M.A.Henning in3_{and was studied, for example in}4, 5, 6, 7, 8_{. A dominating set D}

of L(G) is a regular total dominating set (RTDS) if the induced subgraph < D > has no isolated

vertices and deg(𝑣) = 1, ∀ 𝑣 ∈𝐷. The regular total domination number 𝛾_𝑟𝑡(𝐿 𝐺 ) is the minimum

cardinality of a regular total dominating set. The regular total domination in line graphs was

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domination subdivision numbers of graphs were studied by O. Favaron, H. Karami and S. M.

Sheikholeslami in2_{. On complementary graphs was studied by E. A. Nordhaus and J. W. Gaddum in}_11.

The regular number of graph valued function were studied by M.H.Muddebihal, Abdul Gaffar, and

Shabbir Ahmed in12_{and also developed in}13, 14_.

The following will be useful to prove our results.

Theorem A [2] : If G is a connected graph with p points and q lines and if 𝑏𝑖 is the number of blocks

to which point 𝑣𝑖 belongs in G, then the semitotal block graph 𝑇𝑏(𝐺) has 𝑝𝑖=1𝑏𝑖 + 1 points and q +

𝑏𝑖 𝑝

𝑖=1 lines.

4. RESULTS:

The following result is obvious, hence we omit its proof .

Theorem 1 : For any graph G, 𝑟𝑇𝑏(𝐺) = 1.

if and only if 𝑇𝑏(𝐺) is regular.

Next, we obtain the regular number of a semitotal block graph for a tree.

Theorem 2 : For any non-trivial tree T, 𝑟𝑇_𝑏 𝑇 = ∆ 𝑇 .

Proof : Let u be a vertex of maximum degree ∆ 𝑇 . Let 𝑢1 , 𝑢2 , 𝑢3 , . . . , 𝑢𝑛 be the n vertices adjacent

to u. Since , T is a tree each 𝑢𝑢𝑖 , 𝑖 = 1 , 2 , 3 , . . . , n is a block in T. In 𝑇𝑏(𝑇), let 𝑏1= 𝑢𝑢1 , 𝑏2 = 𝑢𝑢2 , . . . , 𝑏𝑛 = 𝑢𝑢𝑛 such that V[𝑇𝑏(𝑇)] = { 𝑢1 , 𝑢2 , 𝑢3 , . . . , 𝑢𝑛 } ∪ { 𝑢 } ∪ { 𝑏1 , 𝑏2 , 𝑏3 , . . . , 𝑏𝑛 } and each

𝑢𝑢𝑖𝑏𝑖 ; 𝑖 = 1 , 2 , 3 , . . . , n is a block. Let F be a minimum partition of 𝑇𝑏(𝑇) with respect to ∆ 𝑇 ,

where 𝐹1 = { 𝑢𝑢1𝑏1 } 𝐹2 = { 𝑢𝑢2𝑏2 } 𝐹3 = { 𝑢𝑢3𝑏3 } , . . . , 𝐹𝑛 = { 𝑢𝑢𝑛𝑏𝑛 }. Further the remaining

vertices of degree less that ∆ 𝑇 gives partition less than 𝐹𝑛 . Hence 𝑟𝑇𝑏 𝑇 =│𝐹𝑛│= ∆ 𝑇 .

In the following Lemma we establish the regular number of a semitotal block graph for a star.

Lemma 3 : For any star 𝐾1,𝑛 , 𝑟𝑇𝑏 𝐾1,𝑛 = n or 𝑟𝑇𝑏 𝐾1,𝑛 = ∆(𝐾1,𝑛).

Now, we give the exact value of the regular number of a semitotal block graph of a path with p ≥ 2 vertices.

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Proof : Let 𝑃𝑝 : 𝑒1= 𝑣1𝑣2 , 𝑒2= 𝑣2𝑣3 ,𝑒3= 𝑣3𝑣4 , . . . , 𝑒𝑝−2= 𝑣𝑝−2𝑣𝑝−1 ,𝑒𝑝−1 = 𝑣𝑝−1𝑣𝑝 be a path

and every edge in 𝑃𝑝 , 𝑝 ≥ 2 is a block. In 𝑇𝑏(𝑃𝑝) , V[𝑇𝑏(𝑃𝑝)] = V(𝐶𝑝) ∪ { 𝑒1 ,𝑒2 , . . . , 𝑒𝑝 } and

𝑏1= 𝑣1𝑒1𝑣2 , 𝑏2= 𝑣2𝑒2𝑣3 , . . . , 𝑏𝑝−1= 𝑣𝑝−1𝑒𝑝−1𝑣𝑝 . Then clearly 𝑇𝑏(𝑃𝑝) has (p-1) number of blocks

and each block is 2-regular with three vertices. Let F be the minimum regular partition, such that

𝐹1= { 𝑣1𝑒1𝑣2 , 𝑣3𝑒3𝑣4 , . . . , 𝑣𝑝−2𝑒𝑝−2𝑣𝑝−1} and 𝐹2 = { 𝑣2𝑒2𝑣3 , 𝑣4𝑒4𝑣5 , . . . , 𝑣𝑝−1𝑒𝑝−1𝑣𝑝 }

Thus,

𝑟𝑇𝑏 𝑃𝑝 = │{ 𝐹1 , 𝐹2 }│

= 2.

In the following result we establish the regular number of a semitotal block graph of a cycle.

Proposition 5 : For any cycle 𝐶𝑝 , with 𝑝 ≥ 3 , then 𝑟𝑇𝑏 𝐶𝑝 = 𝑝+1

2 ; if 𝑝 is odd.

= 𝑝

2+ 1 ; if 𝑝 is even.

Proof : Let 𝑣1 ,𝑣2 , 𝑣3 , . . . , 𝑣𝑝−1 ,𝑣𝑝 be the vertices of 𝐶𝑝 such that deg 𝑣𝑖= 2 for 1≤𝑖 ≤𝑝 . Let

𝑒1 ,𝑒2 ,𝑒3 , . . . , 𝑒𝑝−1 ,𝑒𝑝 be the edges of 𝐶𝑝 such that 𝑒𝑖= 𝑣𝑖𝑣𝑖+1 for 1 ≤𝑖≤𝑝− 1 and 𝑒𝑝= 𝑣1𝑣𝑝 . In

semitotal block graph of 𝐶𝑝 , 𝑇𝑏(𝐶𝑝) = 𝑊𝑝+𝑣𝑘 , where 𝑣𝑘 is a single tone corresponding to the block 𝐶𝑝 . Now, E( 𝑊𝑝+𝑣𝑘 ) = { 𝑒𝑖 } ∪ { 𝑒𝑝 } ∪ { 𝑒𝑖

′_{} , where}_𝑒

𝑖 = 𝑣𝑖𝑣𝑖+1 ; 1 ≤𝑖≤𝑝− 1 ; 𝑒𝑝= 𝑣1𝑣𝑝 and

𝑒_𝑖′ = 𝑣𝑖𝑣𝑘 , for 1 ≤𝑖≤𝑝 . Then, we consider the following two cases.

Case ( 1 ) : If p is odd, then 𝐹1= { 𝑒1𝑒1′𝑒2′ } 𝐹2= { 𝑒3𝑒3′𝑒4′ } 𝐹3= { 𝑒5𝑒5′𝑒6′ } 𝐹𝑛−1= { 𝑒𝑝′−2 , 𝑒𝑝−2 , 𝑒𝑝−1 ,

𝑒𝑝′ } 𝐹𝑛= { 𝑒2 ,𝑒4 ,𝑒6 , . . . , 𝑒𝑝−3 ,𝑒𝑝 ,𝑒𝑝′−1 } is the minimum regular partition of 𝑇𝑏 𝐶𝑝 .

Thus,

𝑟𝑇𝑏 𝐶𝑝 = │{ 𝐹1 , 𝐹2 ,𝐹3 , . . . , 𝐹𝑛 } │.

= │{ 𝐹1 , 𝐹2 ,𝐹3 , . . . , 𝐹𝑛−1 } │+ 1.

= 𝑝−1 2 + 1

= 𝑝+1 2

Case ( 2 ) : If p is even, then 𝐹1= { 𝑒1𝑒1′𝑒2′ } 𝐹2= { 𝑒3𝑒3′𝑒4′ } 𝐹3= { 𝑒5𝑒5′𝑒6′ } 𝐹𝑛−1 = { 𝑒𝑝−1, 𝑒𝑝′−1 , 𝑒𝑝′ }

and 𝐹𝑛= { 𝑒2 ,𝑒4 ,𝑒6 , . . . , 𝑒𝑝−2 , 𝑒𝑝} is the minimum regular partition of 𝑇𝑏 𝐶𝑝 .

Hence,

𝑟𝑇𝑏 𝐶𝑝 = │{ 𝐹1 , 𝐹2 ,𝐹3 , . . . , 𝐹𝑛 } │.

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= 𝑝 2 + 1 .

Next, we develop the result which gives the relationship between 𝑟𝑇_𝑏 𝐶𝑝 𝑎𝑛𝑑𝑑𝑖𝑎𝑚 𝐶𝑝 .

Theorem 6 : For any cycle 𝐶𝑝 , with 𝑝 ≥ 3 then 𝑟𝑇𝑏 𝐶𝑝 ≤ 𝑝−𝑑𝑖𝑎𝑚 𝐶𝑝 + 1.

Proof : Let 𝐶𝑝 : 𝑣1𝑣2 ,𝑣2𝑣3 ,𝑣3𝑣4 , . . . , 𝑣𝑝−1𝑣𝑝 ,𝑣𝑝𝑣1 be a cycle on diam(𝑐𝑝). Let 𝑒1= 𝑣1𝑣2 , 𝑒2 = 𝑣2𝑣3 ,

𝑒3 = 𝑣3𝑣4 , . . . , 𝑒𝑝−3 = 𝑣𝑝−3𝑣𝑝−2 ,𝑒𝑝−2 = 𝑣𝑝−2𝑣𝑝−1 , 𝑒𝑝−1 = 𝑣𝑝−1𝑣𝑝 , 𝑒𝑝= 𝑣𝑝𝑣1 be the edges of 𝐶𝑝 . In

𝑇𝑏 𝐶𝑝 , V[ 𝑇𝑏(𝐶𝑝)] = p + 1 and E[ 𝑇𝑏(𝐶𝑝)] = { 𝑒𝑖 } ∪ { 𝑒𝑝 } ∪ { 𝑒𝑖′ } such that 𝑒𝑖= 𝑣𝑖𝑣𝑖+1 for 1 ≤𝑖≤𝑝− 1 , 𝑒𝑝= 𝑣𝑝𝑣1 and 𝑒𝑖′ = 𝑣𝑖𝑣𝑘 , for 1 ≤𝑖≤𝑝 . Then clearly 𝐹1= { 𝑒1𝑒1′𝑒2′ }

𝐹2= { 𝑒3𝑒3′𝑒′4 } 𝐹3= { 𝑒5𝑒5′𝑒6′ } , . . . , 𝐹𝑛= { 𝑒2 ,𝑒4 ,𝑒6 , . . . , 𝑒𝑝−2 , 𝑒𝑝} is the minimum regular partition

of 𝑇𝑏 𝐶𝑝 .

Further,

│{ 𝐹1 , 𝐹2 ,𝐹3 , . . . , 𝐹𝑛 }│≤ V 𝐶𝑝 − 𝑝

2+ 1.

│{ 𝐹1 , 𝐹2 ,𝐹3 , . . . , 𝐹𝑛 }│≤ 𝑝 − 𝑝

2+ 1.

Hence,

𝑟𝑇𝑏 𝐶𝑝 ≤ 𝑝−𝑑𝑖𝑎𝑚 𝐶𝑝 + 1.

From Theorem 6, we can generalized the concept to the general graph.

Theorem 7 : For any non trivial graph G, 𝑟𝑇𝑏 𝐺 ≤ 𝑝−𝑑𝑖𝑎𝑚 𝑇𝑏 𝐺 + 2.

Proof : Let 𝐶𝑝 : 𝑣1𝑣2 ,𝑣2𝑣3 ,𝑣3𝑣4 , . . . , 𝑣𝑝−1𝑣𝑝 ,𝑣𝑝𝑣1 be a cycle on 𝑑𝑖𝑎𝑚𝑇𝑏 𝐺 . Let 𝑒1= 𝑣1𝑣2 , 𝑒2 =

𝑣2𝑣3 , 𝑒3 = 𝑣3𝑣4 , . . . , 𝑒𝑝−3 = 𝑣𝑝−3𝑣𝑝−2 ,𝑒𝑝−2 = 𝑣𝑝−2𝑣𝑝−1 , 𝑒𝑝−1= 𝑣𝑝−1𝑣𝑝 , 𝑒𝑝= 𝑣𝑝𝑣1 be the edges of

𝐶𝑝 . In 𝑇𝑏 𝐶𝑝 , V[ 𝑇𝑏(𝐶𝑝)] = p + 1 and E[ 𝑇𝑏(𝐶𝑝)] = { 𝑒𝑖 } ∪ { 𝑒𝑝 } ∪ { 𝑒𝑖′ } such that 𝑒𝑖= 𝑣𝑖𝑣𝑖+1 for 1 ≤𝑖≤𝑝− 1 , 𝑒𝑝= 𝑣𝑝𝑣1 and 𝑒𝑖′ = 𝑣𝑖𝑣𝑘 , for 1 ≤𝑖≤𝑝 .

Then clearly 𝐹1= { 𝑒1𝑒1′𝑒2′ } 𝐹2= { 𝑒3𝑒3′𝑒4′ } 𝐹3= { 𝑒5𝑒5′𝑒6′ } , . . . , 𝐹𝑛= { 𝑒2 ,𝑒4 ,𝑒6 , . . . , 𝑒𝑝−2 , 𝑒𝑝} is

the minimum regular partition of 𝑇𝑏 𝐶𝑝 .

Hence,

𝑟𝑇𝑏 𝐺 ≤ │F │.

𝑟𝑇𝑏 𝐺 ≤ 𝑝−𝑑𝑖𝑎𝑚 𝑇𝑏 𝐺 + 2.

In the next result we obtain a relationship between 𝑟𝑇𝑏 𝐺 and 𝛽1 ∗

𝐺 of a graph.

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Proof : Since each independent edge set partition of G is a regular partition. In 𝑇𝑏(𝐺), by Theorem A

[2] V[𝑇𝑏(𝐺)] > V(G) and E[𝑇𝑏(𝐺)] > E(G).

Hence,

𝑟𝑇_𝑏 𝐺 ≥ 𝛽₁∗(𝐺).

We need the following result to prove the next result.

Theorem 9 : For any graph G, 𝑟𝑇_𝑏 𝐺 ≤𝑞−𝛽₁ 𝐺 + 1.

Proof : Let S be a maximum edge independent set in G. Then 𝐸−𝑆 has at most │ 𝐸−𝑆 │ edge independent sets. Thus, 𝑟𝑇𝑏 𝐺 ≤ │ 𝐸−𝑆 │ + 1.

𝑟𝑇𝑏 𝐺 ≤𝑞−𝛽1 𝐺 + 1.

Now, the following result determines the upper bound on 𝑟𝑇𝑏 𝐺 .

Theorem 10 : For any graph G, 𝑟𝑇_𝑏 𝐺 ≤ 2𝑞−𝑝+ 1.

Proof : By Theorem 9,

𝑟𝑇𝑏 𝐺 ≤𝑞−𝛽1 𝐺 + 1.

Since, 𝛽₁ 𝐺 ≥ 𝛾′ _𝐺_.

𝑟𝑇𝑏 𝐺 ≤𝑞− 𝛾

′ _𝐺_{+ 1.}

Where 𝛾′ _𝐺_{is the edge domination number of G.}

Also, 𝑝−𝑞 ≤ 𝛾′ 𝐺 .

This follows,

𝑟𝑇𝑏 𝐺 ≤𝑞− 𝑝−𝑞 + 1.

𝑟𝑇𝑏 𝐺 ≤𝑞−𝑝+𝑞+ 1.

𝑟𝑇_𝑏 𝐺 ≤ 2𝑞−𝑝+ 1.

In the next result we obtain Nordhaus-Gaddum type result on 𝑟𝑇𝑏 𝐺 .

Theorem 11 : For any non trivial graph G, G ≠ 𝐾𝑝 , 𝑝 ≥ 2 then

𝑟𝑇𝑏 𝐺 + 𝑟𝑇𝑏 𝐺 ≤𝑝 𝑝− 3 + 2 .

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𝑟𝑇𝑏 𝐺 ≤ 2𝑞−𝑝+ 1.

𝑟𝑇_𝑏 𝐺 ≤ 2𝑞 − 𝑝+ 1.

𝑟𝑇_𝑏 𝐺 + 𝑟𝑇_𝑏 𝐺 ≤ 2𝑞−𝑝+ 1 + 2𝑞 − 𝑝+ 1.

𝑟𝑇_𝑏 𝐺 + 𝑟𝑇_𝑏 𝐺 ≤ 2 𝑞+𝑞 − 2 𝑝+ 2.

𝑟𝑇𝑏 𝐺 + 𝑟𝑇𝑏 𝐺 ≤ 2 𝑝

2 −2𝑝+ 2.

𝑟𝑇𝑏 𝐺 + 𝑟𝑇𝑏 𝐺 ≤ 2 𝑝(𝑝−1)

2 − 2𝑝+ 2.

𝑟𝑇𝑏 𝐺 + 𝑟𝑇𝑏 𝐺 ≤ 𝑝(𝑝− 1) − 2 𝑝+ 2.

𝑟𝑇𝑏 𝐺 + 𝑟𝑇𝑏 𝐺 ≤ 𝑝(𝑝− 1 − 2) + 2.

𝑟𝑇𝑏 𝐺 + 𝑟𝑇𝑏 𝐺 ≤𝑝 𝑝− 3 + 2 .

We develop the result in the following theorem by considering a nontrivial tree T with n cutvertices and each n has distinct degrees.

Theorem 12 : For any non trivial tree T with n distinct degrees of cutvertices, 𝑟𝑇𝑏 𝑇 ≥𝑛.

Proof : Suppose G = T has unique cut vertex incident with m number of edges in 𝑟𝑇_𝑏 𝐺 , the

subgraph < 𝑇𝑏 G > =𝑚𝐾3 which gives 𝐹1 , 𝐹2 ,𝐹3 , . . . , 𝐹𝑚 partitions. Hence │𝐹1 , 𝐹2 ,𝐹3 , . . . ,

𝐹𝑚│=𝑚= 𝑟𝑇𝑏 𝑇 . Let 𝑣1 ,𝑣2 , 𝑣3 , . . . , 𝑣𝑛 be the number of cut vertices in T ; deg(𝑣1) ≠ deg(𝑣2) ≠

deg(𝑣3) ≠ , . . . , ≠ deg(𝑣𝑛) and ∀ 𝑣𝑖 ∈ T has degree at least 2. Since each 𝑣𝑖 , 1 ≤𝑖 ≤𝑛 are distinct. Let

{ 𝑒1 ,𝑒2 ,𝑒3 , . . . , 𝑒𝑝 } be the number of edges in T, which are incident to ∀ 𝑣𝑖 1 ≤𝑖 ≤𝑛. Then in

𝑇𝑏(𝑇), the number of edges incident to each 𝑣𝑖 ∈𝐺 generates p number of complete blocks which

are 𝐾3 in 𝑇𝑏(𝑇). In 𝑇𝑏(𝑇) each block is of different regular and is complete. Since T has n- distinct

cut vertices and each block is of same regular, then each block belongs to different partition of

𝑇𝑏(𝑇) such that there exists 𝐹1 ,𝐹2 ,𝐹3 , . . . , 𝐹𝑛 partitions of 𝑇𝑏(𝑇). Hence 𝑟𝑇𝑏 𝑇 ≥ │ { 𝐹1 ,𝐹2 , . . . , 𝐹𝑛 } │ which gives, 𝑟𝑇𝑏 𝑇 ≥𝑛. Hence the proof.

Next, we have the result of regular number of 𝑟𝑇𝑏 𝐺 , where G is unicyclic.

Theorem 13 : Let G be unicyclic graph with 𝐶𝑛 , 𝑛≥ 3

i. if 𝑛≥ ∆(𝐺), then

𝑟𝑇𝑏 𝐺 = 𝑛

2+ 1 ; if n is even.

= 𝑛+1

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Proof : Suppose G is unicyclic graph. Then we consider the following cases.

Case (i) : Suppose G ≅ 𝐶𝑛 , 𝑛≥ 3. Then by Proposition 5, (i) holds.

Case (ii) : Suppose G ≇𝐶𝑛 , 𝑛≥ 3. and G is unicyclic. Then G has exactly one block which is 𝐶𝑛 , 𝑛≥ 3

and remaining blocks of G are edges. Let H = { 𝑣1 ,𝑣2 , 𝑣3 , . . . , 𝑣𝑚} be the set of cut vertices and if

for every 𝑣𝑖∈ 𝐻, deg 𝑣𝑖 <𝑛 of 𝐶𝑛 , 𝑛≥ 3 then in 𝑇𝑏(𝐺) every block is 𝐾3 and the block which is cycle gives a wheel. In edge partition of 𝑇𝑏(𝐺), 𝐹𝑀= { 𝐹1 ,𝐹2 , . . . , 𝐹𝑖} be the partition of the edges

of 𝑇𝑏(𝐺) which does not belongs to 𝐶𝑛 , 𝑛≥ 3 and 𝐹𝑁 = { 𝐹1 ,𝐹2 , . . . , 𝐹𝑗 } be the edge partition of

wheel 𝑊𝑛+1 . Since > deg 𝑣𝑖 , then │𝐹𝑀│ < │𝐹𝑁│. By Proposition 5, again(i) holds. Otherwise if

< deg⁡(𝑣𝑖) , then │𝐹𝑀│ > │𝐹𝑁│ which gives 𝑟𝑇𝑏 𝐺 = ∆ 𝐺 . Hence condition (ii) holds.

Next, we developed the result which gives the relationship between 𝑟𝑇𝑏 𝑇 𝑎𝑛𝑑 𝛾𝑡 𝑇 .

Theorem 14 : For any tree T ,with 𝑝 ≥ 3 vertices and ∆(𝑇) ≥ 𝛾𝑡 𝑇 , then 𝑟𝑇𝑏 𝑇 ≥𝛾𝑡 𝑇 .

Proof : Suppose T =𝑃𝑛 : 𝑣1 ,𝑣2 , 𝑣3 , . . . , 𝑣𝑛 be a path with 𝑛≥ 5 . Then 𝛾𝑡 𝑃𝑛 = 𝑝

2 . Let H = { 𝑏1 ,

𝑏2 , . . . , 𝑏𝑛−1 } be the set of vertices corresponding to the blocks M = { 𝑣1𝑣2 , 𝑣2𝑣3 , . . . , 𝑣𝑛−1𝑣𝑛 }. In

𝑇𝑏(𝑇), V[𝑇𝑏(𝑇)] = {H}∪{M} such that 𝑣𝑖𝑣𝑖+1𝑏𝑖 = 𝐵𝑖 , 𝑖 = 1 , 2 , . . . , 𝑛− 1 in which each 𝐵𝑖 is 𝐾3 . If n is even ≥ 2 , then 𝐹1= {𝐵𝑛−1} and 𝐹2= {𝐵𝑛−2}. If n is odd ≥ 3 , then 𝐹1= {𝐵𝑛−2} and 𝐹2= {𝐵𝑛−1}. These two 𝐹1 and 𝐹2 are the regular partitions of 𝑇𝑏 𝑃𝑛 ,𝑛 ≥ 5. Let D = {𝑣1 ,𝑣2 , 𝑣3 , . . . , 𝑣𝑖} ⊆

V(T) , be the minimal set of vertices which covers all the vertices of V(T) – D in T. Suppose the sub graph <𝐷> has no isolated vertices , then D forms a 𝛾𝑡 set of T. Suppose there exists a vertex

𝑣 ∈𝐷 which is an isolates. Then consider a vertex 𝑤 ∈𝑁(𝑣) such that <𝐷 ∪ {𝑊} > has no isolates. Hence 𝐷 ∪ {𝑊} forms a minimal total dominating set in T. Suppose T =𝑃𝑛 : 𝑣1 ,𝑣2 , 𝑣3 , . . . , 𝑣𝑛 be

a path with 𝑛 ≥ 5 , such that ∆ 𝑇 < 𝛾𝑡 𝑇 . Then the regular partition 𝑇𝑏 𝑃𝑛 = { 𝐹1∪𝐹2 } for

𝑛 ≥ 5. For 𝑃𝑛 ,𝑛 ≥ 5 , │𝐷│ = 𝑛

2≥ │ 𝐹1∪𝐹2│ Which gives 𝛾𝑡 𝑃𝑛 > 𝑟𝑇𝑏 𝑃𝑛 𝑛 ≥ 5. Hence T

≠ 𝑃𝑛 ,𝑛 ≥ 5. Now, we consider a tree T with 𝑝 ≥ 3 with at least one vertex of degree ≥ 3. Suppose

T is a tree with 𝑝 ≥ 3 and ∆(𝑇) ≥𝛾𝑡 𝑇 . Then { 𝐹1 , 𝐹2 , . . . , 𝐹𝑘 } be the regular partition of T such

that │ 𝐹1 , 𝐹2 , . . . , 𝐹𝑘 │= ∆(𝑇). Further D be the minimal dominating set of T such that < 𝐷 > has

no isolates then D is a minimal total dominating set of T. Otherwise we discussed this case as in case of T = 𝑃𝑛 , 𝑛 ≥ 5. Clearly │𝐹1 , 𝐹2 , . . . , 𝐹𝑘│≥

𝑝

2 = │𝐷│ gives 𝑟𝑇𝑏 𝑇 ≥ 𝛾𝑡 𝑇 .

In the next result we discuss a relationship between 𝑟𝑇𝑏 𝑇 𝑎𝑛𝑑 𝑖 𝑇 where 𝑖 is independent

domination.

Theorem 15 : For any non trivial tree T, if ∆ 𝑇 ≥ 𝑖 𝑇 , then 𝑟𝑇𝑏 𝑇 ≥ 𝑖 𝑇 . Converse is also true.

Proof : Let T be a non trivial tree with V = { 𝑣1 , 𝑣2 , 𝑣3 , . . . , 𝑣𝑛 } and deg(𝑣𝑖) = ∆ 𝑇 ,∀ 𝑣𝑖 ∈ 𝑇 , 1 ≤

𝑖 ≤ 𝑛. Then there exists a minimal set S ⊆ V such that N[S] = V(G). Suppose the < S > is disconnected and ∀ 𝑣𝑗 ∈ S is an isolate. Then S is a minimal independent dominating set of T. Suppose │S│≥

∆(𝑇). Then let H = { 𝑣1 , 𝑣2 , 𝑣3 , . . . , 𝑣𝑖 } ⊆ V T be the set of cut vertices such that for any 𝑣𝑖 ∈ H,

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│S│. Hence 𝑟𝑇𝑏 𝑇 ≤ 𝑖(𝑇), a contradiction. Further, if ∆ 𝑇 ≥│S│ then there exist at least a vertex 𝑣 ∈ H such that deg(𝑣) = ∆ 𝑇 . In, 𝑇𝑏(𝑇) the vertex is incident with ∆ 𝑇 number of regular graphs

which are 𝐾3 . Then each 𝐾3 belongs to different partition of 𝑟𝑇𝑏 𝑇 . Hence │∆(𝑇)│ ≥ │S│ gives 𝑟𝑇_𝑏 𝑇 ≥ 𝑖 𝑇 .

5 : Conclusion

We studied the property of our concept by applying to some standard graphs. We also established the regular number of semitotal block graph of some standard graphs. Further we developed the upper bound in terms of minimum edge independence number of G and vertices of G. Also many results established are sharp.

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