split domination number of some special graphssplit domination number of some special graphs 1s....

Split Domination Number of Some Special Graphs 1S. Maheswari and

2S. Meenakshi

1Department of Mathematics, Vels University, Chennai.

2Department of Mathematics, Vels University, Chennai.

Abstract The concept of split domination number was introduced by Kulli and

Janakiram. Let G be a graph with vertex set V and S be the subset of vertex

set V. If every vertex in V-S is adjacent to minimum one vertex of S then S is

said to be a dominating set. If the induced sub graph <V-D> is disconnected

then the graph contains a split dominating set. Let γs(G) be the minimum

cardinality number of split dominating set. In this paper we discussed

about split domination for various types of special graphs in graph theory.

Key Words:Dominating set, split domination number, complementary

edge.

International Journal of Pure and Applied MathematicsVolume 116 No. 24 2017, 103-117ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

103

1. Introduction

In 1958, domination was formalized as a theoretical area in graph theory by

C.Berge. He referred to the domination number as the co-efficient of external

stability and denoted it 𝛽(G). In 1962,Ore was the first to use the term

‘Domination’ number by 𝛿(𝐺) and also he introduced the concept of minimal

and minimum dominating set of vertices in graph.In 1977,Cockayne and

Hedetniemi was introduced the accepted notation 𝛾(G) to denote the domination

number.

The main view of the growth in the number of domination papers is attributed

largely to three factors

1. The diversity of application to real –world and other mathematical

covering of location problems.

2. Wide variety of domination and its related parameters can be defined.

3. The NP completeness of the basic domination problem, it close and

natural relationships to other NP–complete problems and the subsequent

interest in finding polynomials time solutions to domination problems in

special class of graphs

The study of split domination number in graphs has found more development in

the recent years. The size of smallest dominating set is referred to as the

domination number of the graph G has denoted by 𝛾(G).

2. Basic Definitions Definition 2.1: Dominating set

Let G be a graph with vertex set V and edge set E. Let S be the subset of vertex

set V. If every vertex in <V-S> is adjacent to minimum one vertex of S then S

is said to be a dominating set.

Definition 2.2: Dominating number

The size of a smallest dominating set is referred as the dominating number of a

graph G denoted by 𝛾(G).

Definition 2.3: Total Dominating set

A dominating set D is said to be a total dominating set if every vertex in V is

adjacent to some vertex in D. The total domination number of G denoted by

𝛾𝑡 (G) is the minimum cardinality of a total dominating set.

Definition 2.4: Connected Dominating set

A dominating Set D is said to be connected dominating set, if the induced sub

graph <D> is connected. The connected domination number 𝛾𝑐 (G) is the

minimum cardinality of a connected dominating set.

International Journal of Pure and Applied Mathematics Special Issue

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Definition 2.5: Split Dominating set

If the induced sub graph < V – D > is disconnected, then the dominating set D is

called split dominating set. Definition 2.6: Split Domination number

The split dominating number 𝛾𝑠(G) of G is the minimum cardinality of the split

dominating set. Definition 2.7: Non split Domination number

A dominating Set D o f G is a non split dominating set, if the induced sub graph

<V –D> is connected. Non split domination number 𝛾𝑛𝑠 (G) is the minimum

cardinality of a non split dominating set.

3. Important Results of Split Dominating

Set [9]

(i) For a graph G, 1≤ 𝛾𝑠(G) ≤ p-2.

(ii) 𝛾𝑠(𝑃𝑝 ) = [ 𝑝

3 ], p > 2 where 𝑃𝑝 is a path of length p-1.

(iii)𝛾𝑠(𝑘𝑚 ,𝑛 − {𝑒}) = 2.where 𝑘𝑚 ,𝑛 − {𝑒}) is the complement of 𝑘𝑚 ,𝑛 −{𝑒}and {e} is an edge an 𝑘𝑚 ,𝑛 for m,n >1.

(iv) 𝛾𝑠(𝑚𝑘2) = 2m-2, m > 1, where𝑚𝑘2 is the sum m copies of 𝑘2. Theorem 3.1[9]

Let G be a graph. Then 𝛾𝑠 (G) - 1 if and only if there exists only one cut vertex V

in G with degree p-1. Corollary 3.1[9]

If the graph G has no cut vertices, then 𝛾𝑠 (G) ≥ 2.

4. Split Dominating Numbers for Some Special Graphs

Star Graph

Star graph may also described as the only connected graphs in which at most one

vertex has degree greater than one and it has vertices K and edges K+1. Illustration-1

Figure 1: K5 Star graph

This graph shows that there are six vertices namely{𝑉1 , 𝑉2 ,𝑉3 ,𝑉4 ,𝑉5 ,𝑉6}. Here S

is a sub set of G. It has vertex {v6}. If we remove V6 from G, then the graph will


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be disconnected. That is < V-S > = {𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,}. Therefore, < V - S > is

disconnected, therefore the split domination number is 1. i.e., 𝛾𝑠(G) = 1 Bi – Star Graph

Bi- star B (n, n) graph is a graph obtained by joining the apex vertices of two

copies of star 𝐾1,𝑛 .

Illustration-2

Figure 2: Bi- star B (n, n)

This Bi – star graph having the vertices set

V = {𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,𝑉6 ,𝑉7 , 𝑉8 ,𝑉9 ,𝑉10 } and it has a subset S = {u,v}. S is

incident with each and every vertex of G .Then induced sub graph <V-S> is

disconnected so it has domination Number 2. (i.e)., 𝛾𝑠B(n,n) =2.

Tadpole Graph

The (m,n) – Tadpole graph is a special type of graph consisting of a cycle graph

on m(at least3) vertices and a path graph on n vertices, connected with a bridge

denoted by T (m,n)with vertices = m+n and edges = m+n. Illustration-3

Figure 3: T (3, 1) Figure 4: T (4, 1) Figure 5: T (5, 1)

In this tadpole T(3,1) graph having the vertex set {𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,} and it has

subset S .The above graph having the cut vertex 𝑉3 ,thereforeif we remove 𝑉3

from T(3,1) then the graph will be disconnected.

i.e., < V- S> is disconnected therefore it has split domination number 1.

Similarly the graphs T (4, 1), and T (5, 1) having cut vertices so it has a split

domination number 1.Hence T (m, n) = 1. The split domination number for

𝛾𝑠T (m, n) = 1 Wheel Graph

A graph formed from a cycle graph by adding a new vertex adjacent to all the

vertices in the cycle which is dented by W has n vertices and 2(n-1) edges.


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Illustration–4

Figure 6: Wheel graph (W)

Wheel graph contains vertex set V= { 𝑉1 ,𝑉2 , 𝑉3 ,𝑉4 ,} It has the subset namely

S= {v2, v4}.Every vertex of S is incident with< V- S >. Remove S from V, the

graph will be disconnected. So it has a split domination number greater than 2.

i.e., 𝛾𝑆(w) ≥ 2. Banana Tree Graph

An B(n,k) banana tree as defined by Chen et al (1997), is a graph obtained by

connecting one leaf on each of n-copies of an k- star graph with a single root

vertex that distinct from all the stars.

Illustration-5

Figure 7: Banana tree graph (2,3)

Banana graph tree contains vertex set are V = { 𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,𝑉6 ,𝑉7 , 𝑉8 ,𝑉9 ,}

and it has a subset namely S and S has {𝑉2 , 𝑉3} vertices. Every vertex in S is

incident with <V-S>. If S is removed from V, then < V-S > will be disconnected.

Therefore the split domination number for B(2, 3)= 2. Similarly, B (3, 3) =3, B

(4, 3) = 4,…..B (n,k) = n. Therefore, 𝛾𝑆B (n, k) = n. The split dominating number

of a banana tree graph is greater than 2. (i.e)., 𝛾𝑆B (n, k) = n

Butterfly Graph

The butterfly graph (also called the bowtie graph and the hourglass graph) is a

planar undirected graph with 5 vertices and 6 edges. It can be constructed by

joining 2 copies of the cycle graph c3 with a common vertex. Illustration-6

Figure 8: Butterfly graph (D4)

Here the butterfly graph having the vertex set V= {𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,}. Then

subset of the butterfly graph is denoted by S and it has vertex {𝑉1} it is cut vertex


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of Butterfly graph. when the cut vertex is removed, the graph will be

disconnected. That is,< V-S> is disconnected. Therefore it has a split domination

number, the minimum cardinality of the butterfly graph is 1. i.e., 𝛾𝑠(G) = 1

Crown Graph

Crown graph on 2n vertices an undirected graph with two sets of vertices ui and

vj and with an edge from ui to vj whenever i≠ j the crown graph can be viewed as

a complete bipartite graph from which the edges of a perfect matching have been

removed. The split domination number of a crown graph is greater than 2.

Illustration-7

Figure 9: Crown graph (so)

Let {𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,𝑉6 , 𝑉7 , 𝑉8 ,} are the vertices of crown graph.Let S be the

subset of vertex set of crown graph and it is dominated. If every vertex of

S = {𝑉5 ,𝑉6 , 𝑉7 , 𝑉8 } is induced with < V – S>,then S is disconnected. Therefore

the split domination number is greater than 2.i.e., 𝛾𝑠 (𝑠𝑜 ) ≥ 2

Barbell Graph

The n-barbell Graph is a special type of undirected graph consisting of two non

overlapping p – vertex cliques together with a single edge that has a endpoint in

each clique which is denoted by B (p, n) with vertices = 2n. Illustration-8

Figure 10: Barbell graph B (3,2)

Barbell graph having vertices= {𝑉1 , 𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,𝑉6) and it has a subset S whose

vertices are {𝑉2 ,𝑉4 ,}and every vertex of S is incident with <V – S >. When S is

removed from V,<V – S > is disconnected. Similarly, B (4, 2) = 2, B (5, 2) =

2,…

B (m, n) = p-4

5. Split Domination Number of Isomorphic Graph

Graph G(V,E) and G ( V’,E’) are isomorphic, if there is a bijection f: V → V’

such that if (e1,e2) ∈ E then { f(e1),f(e2)} ∈ E’ .


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Illustration-9

Figure 11: Graph G Figure 12: Graph H

D(A) = 1 D(T) = 1

D(B) = 3 D(U) = 3

D(C) = 1 D(V) = 1 Graph (G)

The graph have the vertex set V = {A, B,C,D,E,F,G} and it has a cut vertex D.

we can remove cut vertex from the graph G. Then the graph G will be

disconnected. Therefore, 𝛾𝑠(G) = 1

Graph (H)

The graph H have the vertex set V = {T, U, V, W, X, Y,Z}and it has a cut vertex

v. We can remove cut vertex v from H. Then the graph H will be disconnected.

𝛾𝑠(H) = 1. Here, The split domination number of G = The Split domination

number of H. When a graph is isomorphic the split domination number is also

isomorphic to each other.

6. Split Domination Number

of Complementary Graphs of Some

Special Graphs

The split domination number for complementary graphs of Butterfly graph,

Barbell graph, Tadpole graph, Wheel graph, star graph, bi- star graph do not

exist. Complementary of Crown Graph

The split domination number of complementary of crown graph is greater than 2.

Illustration-10

Figure 13: Complementary of crown graph (sc)

Complementary of crown graph consisting of vertices

V = { 𝑉1 ,𝑉2 , 𝑉3 ,𝑉4 ,𝑉5 ,𝑉6 , 𝑉7 ,𝑉8 ,}


109

Let S be the dominating set of Complementary of Crown graph whose vertex

is{𝑉1 ,𝑉3 ,𝑉6𝑉8 }. Every vertex of S is adjacent with < V-S>.If the vertices of S

are removed from V then the graph gets disconnected. Therefore it has split

domination.

The spit domination number of a complementary of crown be greater than

2.i.e., 𝛾𝑠(𝑠𝑐 ) ≥ 2

7. Split Domination Number of

Kronecker Product of Two Graphs Definition 7.1: Kronecker Product

If G1 and G2 are two simple graphs with their vertex sets 𝑉1: {𝑢1,𝑢2,……} and

𝑉2:{𝑣1,𝑣2,……} respectively. The Kronecker product of these two graphs is

defined to be a graph with its vertex set as 𝑉1x𝑉2 where v1x v2 is the Cartesian

product of the sets 𝑉1,𝑉2and two vertices (𝑢𝑖 ,𝑣𝑗 ),(𝑢𝑘 ,𝑣1) are adjacent iff 𝑢𝑖𝑢𝑘

and 𝑣𝑗𝑣1 are edges in G1& G2 respectively.

Butterfly Graph

The Kronecker product of the butterfly graph can be illustrated below. Illustration - 11

Figure 14: Butterfly graph (D4)

Let the components of the butterfly graph be 𝐺1, 𝐺2are shown below.

𝐺1 𝐺2

Figure 15 Figure 16


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Kronecker Product of a Butterfly Graph

Figure 17

Let the vertex set of Kronecker product of the butterfly graph are

V={ (𝑢1𝑣1),(𝑢1𝑣2),(𝑢1𝑣3),(𝑢2𝑣1),(𝑢2𝑣2),(𝑢2𝑣3),(𝑢3𝑣1),(𝑢3𝑣2),(𝑢3𝑣3)}

Let S be the subset of the Butterfly graph, it has a vertex set

S = {(𝑢1𝑣1), (𝑢1𝑣3) ,(𝑢2𝑣2 ), (𝑢3𝑣1), ( (𝑢3𝑣2 ) }. Here, S is the dominating set of

V. The minimum cardinality, 𝛾𝑠𝐺1(𝑘) 𝐺2 ≥ 2

The split domination number of a kronecker product of butterfly graph is ≥ 2

Star Graph

Let G be the Star graph and it has the components are 𝐺1, 𝐺2 . 𝐺1has the vertex

{𝑢1,} and the 𝐺2 has the vertices {𝑣1,𝑣2 ,𝑣3 ,𝑣4, 𝑣5}. The Kronecker product of

Star graph is illustrated below: Illustration-12

Figure 18

Let the components of star graph be 𝐺1,𝐺2

𝐺1 𝐺2

Figure 19 Figure 20


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Kronecker Product of a Star Graph

Figure 21

Vertices of the Star graph areV= {(𝑢1𝑣1),(𝑢1𝑣2),(𝑢1𝑣3),(𝑢1𝑣5),(𝑢1𝑣6) }

Let S be the subset of the Star graph whose vertices are {(u,v)}. Every vertex of

S is induced with <V – S >.If S is removed from V then the graph will be

disconnected.

Therefore it has a minimum cardinality,𝛾𝑠𝐺1(𝑘) 𝐺2 = 1. The split domination

number of the Kronecker product of the star graph is 1.

Tadpole Graph (3,1)

Let G be the tadpole graph (3, 1) and it has the components are𝐺1, 𝐺2. 𝐺1has the

vertices {𝑢1,𝑢2 ,𝑢3} and the 𝐺2 has the vertices {𝑣1,𝑣2}. The Kronecker product

of tadpole is illustrated below. Illustration-13

Graph G

Figure 22


𝐺1 𝐺2

Figure 23 Figure 24

Kronecker Product of a Tadpole Graph (3,1)

Figure 25


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Tadpole graph (3,1) having the veritces

G1(k)G2 = {𝑢1𝑣1),(𝑢1𝑣2),(𝑢2𝑣1),(𝑢2𝑣2),(𝑢3𝑣1),(𝑢3𝑣2)}

Let S be the subset of the Tadpole graph (3,1).

Let S = {𝑢1𝑣1),(𝑢1𝑣2),(𝑢2𝑣2),(𝑢3𝑣1)}.

Every set of vertex of S is induced with <V – S >. If S is removed from V then

the graph will be disconnected. 𝛾𝑠(𝐺1(𝑘) 𝐺2) ≥ 2.

The split domination number of Kronecker product of Tadpole graph (3, 1) is

greater than or equal to 2.

Tadpole Graph (4,1)

Let G be the tadpole graph (4, 1) and it has the components are𝐺1, 𝐺2.Let 𝐺1has

the vertices { 𝑢1,𝑢2 ,𝑢3, 𝑢4} and 𝐺2 has the vertices {𝑣1, 𝑣2}.Then the Kronecker

product of tadpole is illustrated below.

Illustration-14

G

Figure 26


𝐺1 𝐺2

Figure 27 Figure 28

Kronecker Product of a Tadpole Graph (4, 1)

Figure 29


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Let the vertex set of Kronecker product of tadpole graph (4,1) be G1(k)G2 =

{(𝑢1𝑣1),(𝑢1𝑣2),(𝑢2𝑣1)((𝑢2𝑣2),(𝑢2𝑣3),(𝑢3𝑣1),(𝑢3𝑣2),(𝑢4𝑣1), ),(𝑢4𝑣2)}

Let S be the subset of the Kronecker product of tadpole graph (4,1). It is a

dominating set and every vertex of S can be induced with < V – S>.So it has a

minimum cardinality.

S = {(𝑢1𝑣1),(𝑢2𝑣1),(𝑢3𝑣1),(𝑢4𝑣1),(𝑢3𝑣2)}

Therefore, < V – S>is disconnected. Here, 𝛾𝑠𝐺1(𝑘) 𝐺2 ≥ 2.

The split domination number of the Kronecker product of tadpole graph (4,1)

is greater than or equal to 2. Wheel Graph

Let G be the Wheel graph and it has the components are𝐺1, 𝐺2.Let 𝐺1has the

vertices {𝑢1,𝑢2 ,𝑢3} and the 𝐺2 has the vertices {𝑣1, 𝑣2, 𝑣3,𝑣4}.Then the

Kronecker product of tadpole is illustrated below

G

Figure 30: Wheel graph (W)


𝐺1 𝐺2

Figure 31 Figure 32

Kronecker Product of a Star Graph

Figure 32


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Let the vertex set of the kronecker product of a wheel graph beV(G1(k)G2)=

{(𝑢1𝑣1),(𝑢1𝑣2),(𝑢1𝑣3),(𝑢1𝑣4),(𝑢2𝑣1),(𝑢2𝑣2),(𝑢2𝑣3),(𝑢2𝑣4),(𝑢3𝑣1),(𝑢3𝑣2),(𝑢3𝑣3

), (𝑢3𝑣4)}.

Let S be the Subset of the Kronecker product of a star graph and its vertices are

S = {(𝑢1𝑣1), (𝑢2𝑣1), (𝑢3𝑣1)}. Every vertex of S can be induced with < V – S >.

If S is removed from V then the graph will be disconnected. So it has a minimum

cardinality number greater than 2. 𝑖. 𝑒. ,𝛾𝑠(𝐺1 𝑘 𝐺2) ≥ 2.

8. The table given below represents the split domination number for special graphs and complementary and the Kronecker products of the graph

S.No Name of

special graph

Split domination number for

special graph


complementary


Kronecker product of graphs

1 Star graph 1 Does not exist 1

2 Bi-star graph 2 Does not exist -

3 Tadpole graph 1 Does not exist ≥2

4 Wheel graph ≥2 Does not exist ≥2

5 Banana tree

graph

n Does not exist -

6 Butterfly graph 1 Does not exist ≥2

7 Barbell graph p-4 ≥2 -

9. Conclusion

In this paper the author discussed about the split domination number for some

special graphs and also found the split domination number for complementary of

some special graphs, Kronecker product of special graphs.

References

[1] Harneys T.W., Hedetniemi S.T., Slater P.J., Domination in Graphs-Advanced Topics, Marcel Dekker, Inc., New York (1998).

[2] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals of domination in graphs, Marcel Decker Inc., New York (1998).

[3] Joseph J.P., Arumugam S., Domination and connectivity in graphs, International Journal of Management and Systems, volume (1992), 233-236.

[4] Kulli V.R., Janakiram B., The split domination number of a graph. Graph theory notes, New York (1997).

[5] Kulli V.R., Janagiram B., The non split domination number of a graph”, Indian Journal. Pure Applied Mathematics 31(4) (2000), 545-550.


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[6] Kulli V.R., Sigarkanti S.C., Soner N.D, Entire domination in gaphs. Advances in Graph theory, Vishwa International Publications (1991), 237-243.

[7] Sivagnanam C., Kulaindaivel M.P., Complementary Connected Domination Number and Connectivity of a Graph, General Maths Notes (2015), 27-35.

[8] Tamizh Chelvam T., Jaya Prasad B., Complementary Connected Domination Number, International Journal, Management systems 18(2) (2002), 147-154.

[9] Chelvam T.T., Chellathurai S.R., A note on split domination number of a graph, Journal of Discrete Mathematical Sciences and Cryptography 12(2) (2009), 179-186.

[10] Haynes T.W., Hedetniemi S., Slater P., Fundamentals of domination in graphs, CRC Press (1998).


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