Cmjv04i01p0069

J. Comp. & Math. Sci. Vol.4 (1), 69-74 (2013)

Consecutive Adjacent Domination Number in Semigraphs D. K. THAKKAR1 and A. A. PRAJAPATI2 1

Department of Mathematics, Saurashtra University Campus, Rajkot-360 005, INDIA. 2 Mathematics Department, L. D. College of Engineering, Ahmedabad-380 015, INDIA. (Received on: February 17, 2013) ABSTRACT In this paper we consider consecutive adjacent domination number in semigraphs. In particular, we prove some conditions under which the consecutive adjacent domination number of a semigraph increases or decreases.

Keywords: Semigraph, Consecutive adjacent domination, consecutive adjacent domination number, consecutive adjacent dominating set.

AMS Subject Classification: 05C69 and 05C99.

1. INTRODUCTION

2. PRELIMINARIES

Semigraph is a combinatorial structure closely related to a graph. In a semigraph an edge will contain a more than two vertices in general. In fact an edge in semigraph is an n-tuples for n≼2. We may define domination in semigraph. However we may have variants of domination in semigraph.

Definition 2.1: Semigraph1

In this paper we consider one such variant of domination called consecutive adjacent domination in semigraph. In fact we investigate the conditions under which the consecutive adjacent domination number of a semigraph increases or decreases.

a) Any two edges have at most one vertex in common. b) Two edges ( , , ..., ) and ( , , ..., ) are considered to be equal if and only if 1. and

A semigraph G is a pair (V, X) where V is a nonempty set whose elements are called vertices of G, and X is a set of ntuples, called edges of G, of distinct vertices, for various n≼2, satisfying the following conditions:

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

70

D. K. Thakkar, et al., J. Comp. & Math. Sci. Vol.4 (1), 69-74 (2013)

2. Either for 1 , or for 1 .

Thus, the edge E = ( , , ..., ) is the same as ( , , ‌ . and are called the end vertices of an edge E and , , ... , are called the middle vertices of E. Example 2.2: Let G = ( V, X ) be a semigraph where V = {1,2,3,4,5,6} and X = {(1,2,3), (3,4 ), ( 4,5 ), (5,6,3 ), (1,6 ), ( 6,4 ) }. In G 1 and 3 are end vertices of an edge (1, 2, 3) and 2 is middle vertex of this edge.

1 5 2

6

3

4

where 1 ‌ or 1 ‌ . We say that the subedge

is induced by te set of vertices , , ‌ , .

Definition: 2.4: Subsemigraphs:1 A semigraph

,

is said be subsemigraph of semigraph , if

and the edges in are subedges of G. Definition: 2.5: dominating set:2

Consecutive

adjacent

Let G = (V, X) be a semigraph. A subset S of V is said to be consecutive adjacent dominating set if for every vertex , there exist a vertex in S such that and are consecutive adjacent in a semigraph G.

Figure-1

Definition: 2.6: Minimal adjacent dominating set: If E = ( , , ..., ) is an edge in a semigraph then for any , and are said to be consecutive adjacent vertices of the edge E. In above semigraph 1 and 2 are consecutive adjacent vertices of the edge (1, 2, 3).

Two vertices and are said to be consecutive adjacent, if they are consecutive adjacent in some edge E. If is vertex in semigraph then

.

Definition: 2.3: Subedge:1 subedge of an edge , , ‌ , is a -tuple భ , ఎ , ‌ , ೖ , A

consecutive

A consecutive adjacent dominating set S is said to be minimal consecutive adjacent dominating set if every vertex , is not a consecutive adjacent dominating set.

Definition: 2.7: Minimum consecutive adjacent dominating set: A consecutive adjacent dominating set with minimum cardinality is said to be a minimum consecutive adjacent dominating set. This set is denoted as - set.

The cardinality of such a set is called consecutive adjacent domination number of G and is denoted as .

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

D. K. Thakkar, et al., J. Comp. & Math. Sci. Vol.4 (1), 69-74 (2013)

Definition: 2.8: Consecutive private neighbourhood:

adjacent

Let G be a semigraph, S be a set of vertices of G and , then the consecutive adjacent private neighbourhood of with respect to / . This set is denoted as ! , .

Also then denotes the subsemigraph induced by the vertices of

.

We introduce the following three notations:

:

:

: #

We may note that

$

$ and these three sets are mutually disjoint. In this paper we assume that all semigraphs S are simple.

3. MAIN RESULTS We first prove the following simple results.

Theorem: 3.1 The following three statements are equivalent for semigraph G and ,

71

1 # 3 Let S be a minimum consecutive adjacent dominating set of . In any case $ will be a consecutive adjacent dominating set of G. Therefore,

| $ | | | ' 1. Since, " #ŕŻ–ŕŻ”ŕŹż , ௖௔ | | 1 | |. Thus, is a -set of .

3 1 ௖௔ ௖௔ and hence

.â–ˆ Theorem: 3.2 If G is a semigraph and is a

vertex of G such that then belongs to every minimum -set of G. Proof: If there is a minimum ௖௔ - set such that ( then it is obvious that is a consecutive adjacent dominating set of , this implies that ௖௔ % | ଴ | ௖௔ this means that & #ŕŻ–ŕŻ”ŕŹž , a contradiction. â–ˆ Next we prove a necessary and sufficient

. condition under which Theorem: 3.3 If G is a semigraph and

then if and only if the following conditions hold.

1) .

2)

1.

3) For every set of , $ is a set of G.

1) is not consecutive isolated in G. 2) There is no subset S of such that | | % ௖௔ , S is a minimum consecutive adjacent dominating set of with $ is a consecutive adjacent dominating set in G.

Proof: 1 2 Let ଵ be a ௖௔ -set .

Then $ is a consecutive adjacent dominating set of .Therefore, ௖௔ | | | ଵ | 1. Thus, ௖௔ ௖௔ 1.

Hence,

1.

2 1 The statement implies that ௖௔ ௖௔ .

Proof: Suppose is consecutive adjacent isolated vertex in G. If S is any -set of G, then must be in S. Then is a consecutive adjacent dominating set of ௖௔ % | | | | .Therefore

௖௔ , this means that ( , a contradiction.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

72

D. K. Thakkar, et al., J. Comp. & Math. Sci. Vol.4 (1), 69-74 (2013)

To prove the second conditions suppose there is such a set S. This implies that

| | . This means

that . Again a contradiction.

Now we will prove that (

and ( .This will implies that .

Suppose , Let S be - set of . If S is a consecutive adjacent dominating set of then so is $ . This contradicts second condition.

If is not consecutive adjacent dominating set in then $ will be a consecutive adjacent dominating set of . This is again a contradiction.

Suppose . Let S be a set of then | | 1 . This set S cannot be a consecutive adjacent dominating set of G. Clearly, $ is a consecutive adjacent dominating set of .

Thus second condition is violated.

Hence ( .â–ˆ

4. ) SEMIGRAPH Of G: Now we consider another semigraph whose vertex set same as the given semigraph and the edges are those edges of G which do not contain the vertex or contained the vertex as an end vertex. We called this semigraph to be semigraph of G. We denote this semigraph as .

Example 4.1: Let G be a semigraph whose vertex set # 1,2,3,4,5,6 and * 1,2,3 , 3,4 , 4,5 , 5,6,3 , 1,6 , 6,4 we select the vertex 6, the -semigraph has the vertex set # 1,2,3,4,5,6 and the edge set * 1,2,3 , 3,4 , 4,5 , 1,6 , 6,4

1

5

2

6

3

4 -semigraph

Notation: (1) For , (2)

: .

For ,

:

.

We introduce the following notations for the purpose of change in the domination number when the vertex is removed from the semigraph . #ŕŻ–ŕŻ”ŕŹ´ + " # ொ : ௖௔ + ௖௔ #ŕŻ–ŕŻ”ŕŹż + " # ொ : ௖௔ + ௖௔ #ŕŻ–ŕŻ”ŕŹž + " # ொ : ௖௔ + ௖௔

We may note that

$ $ and these three sets are mutually disjoint.

Theorem: 4.2 if and only if the following three conditions are satisfied.

1) is not a consecutive isolated vertex in .

2) belongs to every - set of .

3) There is no subset of

with | | ! and dominating

.

Proof: Suppose

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

D. K. Thakkar, et al., J. Comp. & Math. Sci. Vol.4 (1), 69-74 (2013)

+ (1) If is consecutive isolated in , then for any - set of of , .

Now consider . If , is any vertex of , which is not in , then since, is consecutive adjacent dominating set of , , must be consecutive adjacent to some vertex - of in . This vertex - cannot be equal to . As is consecutive isolated in ொ .Thus, . is consecutive adjacent to some vertex of .

Thus, is consecutive adjacent dominating set of .This implies that | | | |

.

Hence, ( , a contradiction. Thus, is consecutive non isolated in .

(2) Suppose there is a minimum consecutive adjacent dominating set of such that ( . Let , be a vertex in such that , ( .

Now , is consecutive adjacent to some vertex - in and / 0 . As & . Thus, is consecutive adjacent dominating set of ொ .This implies that ௖௔ ொ % | | | | ௖௔ ொ .

This again contradicts the fact that " #ŕŻ–ŕŻ”ŕŹž .

(3) If there is a set . such that | | and is a consecutive adjacent dominating set of then

| | and again we have a contradiction.

Thus, condition (3) is also satisfied. / Conversely, we wish to prove that

.

First suppose that " #ŕŻ–ŕŻ”ŕŹ´ . Let be a ௖௔ - set of ொ . We may not that if , is any vertex not in and . 0 , then , is consecutive adjacent to some vertex / in in ொ , because appears only as an end vertex in every edge in which it is present.

73

Hence, if is consecutive adjacent to some vertex of in , then is a minimum consecutive adjacent dominating set in not containing the vertex , which contradicts condition (2).

Suppose is not consecutive adjacent to any vertex of , then . and is consecutive adjacent dominating set of with | | . This contradicts condition (3).

Thus, we have a contradiction if .

Suppose . Let be a minimum consecutive adjacent dominating set of . If is not consecutive adjacent to any vertex of , then .

, | | and is a consecutive adjacent dominating set of , which contradicts condition (3) again.

If is consecutive adjacent to some vertex of , then it will imply that is a consecutive adjacent dominating set of with | | .

This is a contradiction. Thus, we have a contradiction

in both the cases if .

Thus, we have proved that ( , (

and hence .â–ˆ We may state the following theorem without prove.

Theorem: 4.3 Let be a consecutive adjacent dominating set in . Then is a minimal consecutive adjacent set in , if and only if for every 0 , ! 0, 1 2.

Now we prove following Lemma.

Lemma: 4.4 If , then

1.

Proof: Let be a minimum consecutive adjacent dominating set of .

Now cannot be consecutive adjacent to any vertex of in . Otherwise, would

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

74

D. K. Thakkar, et al., J. Comp. & Math. Sci. Vol.4 (1), 69-74 (2013)

be a consecutive adjacent dominating set of with | | , which is a contradiction.

Now consider a set $ , then obviously is minimum consecutive adjacent dominating set of ொ .Thus ௖௔ ொ | | ௖௔ ொ 1.

Hence,

1.â–ˆ

Theorem: 4.5 if and only if there is a - set of such that and consecutive private neighbourhood of with respect to .

. Let be a Proof: Suppose minimum consecutive adjacent dominating set of . As proved in the lemma 3.6 cannot be a consecutive adjacent to any vertex of .

Let $ , then is a minimum consecutive adjacent dominating set of and is not consecutive adjacent to any vertex of . So belong to ! , . If 0 1 and 0 ( , then 0 is a vertex of out side . Hence 0 is consecutive adjacent to some vertex 3 of . If 0 is consecutive adjacent to then it will imply that 0 is consecutive adjacent to two distinct vertices 3 and of .Thus, 0 ( ! , .

Hence, ! 0, .

Conversely, Suppose for some minimum consecutive adjacent dominating set of such that ! 0, .

Now consider the set .

If 0 is any vertex of , which is not

in , then if it is consecutive adjacent to then it must be a consecutive adjacent to some other vertex - of , that is 0 is consecutive adjacent to some vertex of - of . If 0 is not consecutive adjacent to , then 0 must be consecutive adjacent to some vertex 4 of . Thus, in any case 0 is consecutive adjacent to some vertex of .

Hence, is a consecutive adjacent dominating set of Thus, | | | |

â–ˆ Hence, REFERENCES 1. E. Sampthkumarachar With contributions from: L. Pushpalatha, B. Y. Bhave, C. M. Despande, Semigraphs and their Applications.

2. S. S. Kamath and R. S. Baht, Domination in semigraphs, Preprint submitted to Elsevier Science, 12 September (2003).

3. Shyam S. Kamath and Saroja R. Hebbar, Domination In Critical Semigraphs, Domination In Semigraphs (Part – III), Lecture Notes on National Workshop on Semigraphs, College of Engineering, Pune, (2010).

4. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination In Graphs Advanced Topics, Marcel Dekker, Inc, (1998).

5. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamental Domination In Graphs, Marcel Dekker, Inc, (1998).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)