On the k-Metro Domination Number of Paths

Vol. 14, No. 3, 2017, 593-600

ISSN: 2279-087X (P), 2279-0888(online) Published on 7 December 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/apam.v14n3a26

593

Annals of

On the k-Metro Domination Number of Paths

Lakshminarayana S1 and Vishukumar M2

1

School of Applied Sciences, REVA University, Bangalore-560064, India. E-mail: narayan_sl@reva.edu.in

2

School of Applied Sciences, REVA University, Bangalore-560064, India. E-mail: vishukumarm@reva.edu.in

Received 2 November 2017; accepted 15 November 2017

Abstract. A dominating set D of a graph = (, ) is called metro dominating set of G. If for every pair of vertices u, v there exists a vertex w in D such that ( , ) ≠ (, ), The k-metro domination number of a path (), is the order of a smallest k-dominating set of which resolves as a metric set. In this paper, we calculate the k-metro domination number of paths.

Keywords: Domination, metric dimension, metro domination. AMS Mathematics Subject Classification (2010): 05C56, 05C38 1. Introduction

Let (, ) be a graph. A subset of vertices ⊆ is called a dominating set (- set) if every vertex in − adjacent to at least one vertex in D [4].

The minimum cardinality of a dominating set is called the domination number of the graph G and is denoted by ().[4].

The metric dimension of a graph G is denoted by () is defined as the cardinality of a minimal subset ⊆ having the property that for each pair of vertices u, v in G there exists a vertex w in S such that ( , ) ≠ (, )The coordinate of each vertex of V (G) with respect of each landmark belong to S is defined as usual with ith component of v as ( , ) for each i and is of dimension ()[2].

Metro domination number introduced by Sooryanarayana and Raghunath [5]. Fink and Jacobson [6, 7] in 1985 introduced the concept of multiple domination. A subset D of V (G)is k-dominating in G if every vertex of − has at least k neighbor’s in D. The cardinality of minimum k- dominating set is called k- domination number (). A dominating set D of a graph G(V;E) is called metro dominating set of G if for each pair of vertices u; v there exists a vertex w in D such that ( , ) ≠ (, ).

2. Our results

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Proof: Let _$, _%, _&, … … … … . . be the vertices of the path . Let D be the minimum 2-dominating set of . Let ( = − , Now each ∈ ( is either adjacent to any of the vertex D or atmost at distance two from atleast one of the vertex of D. Any vertex _* ∈ , will dominates at most 5 vertices including itself. Since the metric dimension of the Path () = 1, as in [5], D itself serve as a metric set.

Thus () ≥ !

"# (i) To prove () ≤ !

"#, We define a set D as follows

Case 1: For = 5- + 6, - ≥ 1,

= 0"*1&; 0 ≤ 4 ≤ 56%_" 78 ∪

Case 2: For = 5- + 7, - ≥ 1,

= ;"*1&; 0 ≤ 4 ≤ < − 3_{5 >? ∪}

Case 3: For = 5- + 8, - ≥ 1,

= ;"*1&; 0 ≤ 4 ≤ < − 4_{5 >? ∪}

Case 4: For = 5- + 9, - ≥ 1, = 0"*1&; 0 ≤ 4 ≤ 5_{578 ∪}

Case 5: For = 5- + 10, - ≥ 1,

= ;"*1&; 0 ≤ 4 ≤ < − 1_{5 >? ∪}

We note that D is a 2-dominating set for , and also D will serves as a metric set of .

Thus () ≤ !

"# (ii)

From (i) and (ii)

() = !_"#.

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Theorem 2.2. For ≥ 8 , _C() = ! D#.

Proof: Let _$, _%, _&, … … … … . . be the vertices of the path . Let D be the minimum 3-dominating set of . Let ( = − , Now each ∈ ( is either adjacent to any of the vertex D are atleast at the distance three from atleast one of the vertex of D. Any vertex _* ∈ , will dominates at most 7 vertices including itself, D also serves as a metric set.

Thus _C() ≥ !_D# (i) To prove _C() ≤ !_D#,

We define a set D as follows

Case 1: = 7- + 1, 7- + 2, 7- + 3, - ≥ 1

= 0D*6&; 1 ≤ 4 ≤ 5_D78 ∪

Case 2: = 7- + 4, 7- + 5, 7- + 6, 7- + 7, - ≥ 1

= 0D*6&; 1 ≤ 4 ≤ 51&_D 78.

We note that D is a 3-dominating set for , and also D will serves as a metric set of .

Thus _C() ≤ !_D# (ii) From (i) and (ii)

C() = !

D#.

Case 1: = 7- + 1, 7- + 2, 7- + 3, - ≥ 1,

3

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G#.

Proof: Let _$, _%, _&, … … … … . . be the vertices of the path . Let D be the minimum 4-dominating set of . Let ( = − , Now each ∈ ( is either adjacent to any of the vertex D are atleast at the distance four from atleast one of the vertex of D. Any vertex _* ∈ , will dominates at most 9 vertices including itself D also serves as a metric set.

Thus _F() ≥ !

G# (i) To prove _F() ≤ !_G#,

We define a set D as follows

Case 1: = 9- + 1, 9- + 2, 9- + 3, 9- + 4, 9- + 5, - ≥ 1

= 0G*6H; 1 ≤ 4 ≤ 5_G78 ∪

Case 2: = 9- + 6, 9- + 7, 9- + 8, 9- + 9, - ≥ 1

= 0G*6H; 1 ≤ 4 ≤ 51&_G 78.

We note that D is a 4-dominating set for , and also D will serves as a metric set of .

Thus _F() ≤ !_G# (ii) From (i) and (ii)

F() = !

G#.

Case 1: = 9- + 1, 9- + 2, 9- + 3, 9- + 4, 9- + 5 , - ≥ 1,

4 (1 3 ) 3 (1 2 ) 2 (1 1 ) 1 (1 0 ) 0

9 ₁8

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Theorem 2.4. For ≥ 12, _I() = ! $$#.

Proof: Let _$, _%, _&, … … … … . . be the vertices of the path. Let D be the minimum 5-dominating set of . Let ( = − , Now each ∈ ( is either adjacent to any of the vertex D are atleast at the distance five from atleast one of the vertex of D. Any vertex _*∈ , will dominates at most 11 vertices including _*, D also serves as a metric set of .

Thus _I() ≥ !

$$# (i) To prove _I() ≤ !

$$#, We define a set D as follows

Case 1:

= 11- + 1, 11- + 2, 11- + 3, 11- + 4, 11- + 5, 11- + 6, 11- + 7 , - ≥ 1, = 0$$*6"; 1 ≤ 4 ≤ 5_$$78 ∪

Case 2: = 11- + 8, 11- + 9, 11- + 10, 11- + 11, - ≥ 1

= 0$$*6"; 1 ≤ 4 ≤ 51&_$$78.

We note that D is a 5-dominating set for , and also D will serves as a metric set of .

Thus _I() ≤ !

$$# (ii) From (i) and (ii)

I() = !_$$#. Case 1:

= 11- + 1, 11- + 2, 11- + 3, 11- + 4, 11- + 5, 11- + 6, 11- + 7, - ≥ 1,

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Theorem 2.5. For ≥ 2- + 6, - ≥ 1,

() = !_%J1& # , 4 ≥ 2, K ≥ 1.

Proof follows from generalization of theorem1, theorem2, theorem3 and theorem4.

Acknowledgements. The authors would like to thank the referees for their valuable suggestions and very much thankful to the Management of Reva University, Rukmini Knowledge Park, Kattigenahalli, yelahanka Bengaluru-64.

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