methodology for finding location domination number of ......neyveli, tamil nadu, india. abstract a...

Methodology for finding Location Domination Number of Complement of a

Graph

G. Rajasekar

Associate Professor, Department of Mathematics,

Jawahar Science College,

Neyveli, Tamil Nadu, India.

K. Nagarajan

Research Scholar, Department of Mathematics,

Jawahar Science College,

Neyveli, Tamil Nadu, India.

Abstract

A dominating set S which locate all vertices uniquely in the

set V G S is a locating dominating set. This paper presents

the procedure for finding the location domination number of

Complement of any graph.

Keywords: Locating domination set, 1 -vertex non locating

dominating graph, Wholly located dominated graph,

Complement of graph.

Introduction The Oystein Ore [1] defined that the dominating set of a graph

G is a subset 𝑆 of the vertex set ( )V G such that all vertices

in the set V G S is adjacent to atleast one vertex in S .

Minimal cardinality of dominating set is known as domination

number and it is denoted by G .

For simple graph , G V G E G , open neighbourhood

GN v of a vertex ( )v V G is

| GN v u V G uv E G

And

| GN v u V G uv E G v

is the closed neighbourhood of v V G [2].

Slater [3,4] defined the locating dominating set ( LD -set) of

the graph G is a dominating set S with ( ) ( ),G GS v S w for

any , v w V G S , where ( )G GS v N v S . The

minimum cardinality of a LD -set in G is called the location-

domination number of G and it is denoted by ( )RD G . An

LD -set with ( )RD G elements is called as a referencing-

dominating set or an RD -set.

Hernando et al [5] found the Nordhaus-Gaddum bounds for

location domination number and studied the relation between

the location domination number of bipartile graph, block

cactus and their complements [6,7].

Preliminaries

Definition 2.1 ([8]) Let G be any graph with RD -set S . If

G has a vertex v V G S , such that S v S then the

vertex v is said to be wholly located dominated vertex with

respect to the RD - set S .

Definition 2.2 ([8]) Let S be the collection of all RD -set of

G . The graph G is said to be wholly located dominated

graph if for every RD -set SS there exist a wholly located

dominated vertex.

Definition 2.3 ([9]) For the given graph G , a set S V G

is said to be 1 -vertex non locating dominating set if it satisfies

the following conditions

(i) With respect to some vertex v V G , S must be the

RD -set of G v

(ii) GN v S

(iii) S RD G

(iv) Any set with cardinality less than S will not satisfies

(i), (ii) and (iii)

(v) If S RD G with S u S for some vertex

u V G S then for every RD -set S of the graph

G there exist some vertex w V G S such that

S w S .

Graphs for which the 1 -vertex non location domination set

can be found are said to be1 -vertex non locating dominating

graph.

Remark 2.1 ([9]) 1 -vertex non location domination set for the

given graph can be obtained from the RD -set S of the graph

G by removing a vertex from S or interchanging a vertex

from S with vertex from V G S . Hence cardinality of S

can be either RD G or 1RD G .

Theorem 2.1 ([5]) If S is an LD -set of a graph G then S is

also an LD -set of G , unless there exist a vertex \w V S

such that GS N w , in which case S w is an LD -set

of G .

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 3, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com

Page 33 of 36

Theorem 2.2 ([5]) For every graph G ,

1 1RD G RD G RD G .

Location Domination Number of Complement of a Graph

Lemma 3.1: Suppose G has a 1 -vertex non locating

dominating set S then S or S v will be the LD -set of

G where vertex v has the characteristic that S v S .

Proof: By definition of 1 -vertex non locating dominating set

there exist a vertex u V G such that GN u S and

for all 1 2,v v V G u S ,

1 1

2

2

G

G

S v N v S

N v S

S v

With respect to the graph G , for any vertex w V G S ,

GG GN w S N w S S N w S

For any vertex 1 2,v v V G u S ,

1 1

2

2

GG

G

G

N v S S N v S

S N v S

N v S

as 1 2G GN v S N v S , 1 2,v v V G u S .

And for the vertex u ,

GG GN u S N u S S N u S

S S

Hence S will be the LD -set of G unless G has a vertex v

such that S v S . In that case S v will be the LD -

set of G .

Lemma 3.2: For any graph G , 1RD G RD G iff G is

a 1 -vertex non locating dominating graph with a 1 -vertex non

locating dominating set S such that for some vertex

1v V G , 1GN v S with 1S RD G and

S v S for all 1v V G v S .

Proof: Let S be the RD -set of G with cardinality

1RD G . Then by definition of LD -set, S cannot be the

LD -set of G . This is due to one of the following

circumstances

(i) For some vertices 1 2,v v V G S ,

1 2G GN v S N v S

(ii) 1GN v S for some vertex 1v V G

If 1 2G GN v S N v S then

1 1

1

2

2

2

GG

G

G

G

G

N v S V G N v S

V G N v S

V G N v S

V G N v S

N v S

Therefore S cannot be the RD -set of G . But this is a

contradiction to our assumption, therefore 1GN v S

for some vertex 1v V G .

i.e. S locate and dominate all vertices 1v V G v S

with 1S RD G and 1GN v S . Hence S is an 1 -

vertex non locating dominating set of G with cardinality

1RD G .

Since S is the RD -set of G , GN v S for all

v V G S , for any vertex v V G S ,

G G

G

G

G

N v S V G N v S

V G S N v S

S N v S

S N v S

S

i.e. for the graph G , S is a 1 -vertex non locating dominating

set with 1 1G GS v N v S and

G GS v N v S S for all 1v V G v S .

Conversely, let us assume that G has a 1 -vertex non locating

dominating set S which satisfies the hypothesis of the

theorem. Then by Lemma 3.1, S is the LD -set of G . But as

1S RD G and by Theorem 2.2, S must be the RD -

set of G .

Lemma 3.3: Suppose G is a 1 -vertex non locating

dominating graph with 1RD G RD G then all 1 -vertex

non locating dominating set S of the graph G have the

following properties:

(i) S RD G

(ii) S v S for some vertex v V G S

Proof: By properties of S , 1RD G S RD G .

Suppose 1S RD G then RD G can be atmost

RD G provided there exist a vertex v V G S such

that S v S . Therefore S cannot be 1RD G and

hence S RD G .

Now suppose S RD G and S v S for all

v V G S then by Lemma 3.1, S will be the LD -set of


Page 34 of 36

G with cardinality RD G . This contradict the theorem’s

hypothesis, hence there exist a vertex v V G S such that

S v S and S v is the RD -set of G .

Theorem 3.1: Let G be a 1 -vertex non locating dominating

graph. Then G has a RD -set which contains the 1 -vertex

non locating dominating set of G .

Proof: By Theorem 2.2, RD G can be equal to one of the

following values 1RD G , RD G or 1RD G .

Case 1: 1RD G RD G

By Lemma 3.2, G has a 1 -vertex non locating dominating set

S such that 1S RD G and S v S for all

v V G S which will be the RD -set of G .

Case 2: 1RD G RD G

By Lemma 3.3, G has a 1 -vertex non locating dominating set

S such that S RD G and S v S for some

v V G S . And RD -set of G is S v .

Case 3: RD G RD G

Let S be a 1 -vertex non locating dominating set of G and

by definition 1RD G S RD G .

Suppose if 1S RD G then there exist a vertex

v V G S such that S v S otherwise by

Lemma 3.1, S would be the LD -set of G with cardinality

less than our assumption. Hence S v is the RD -set of

G .

If S RD G then S v S for all v V G S

otherwise by definition of S , for every RD -set S of the

graph G there exist a vertex v V G S such that

S v S . Then none of RD -set of G will be the RD -set of

G and therefore RD G RD G . This doesn’t go along

with our assumption. Hence S is the RD -set of G .

From Case 1, 2 and 3, we can conclude that if G is a 1 -vertex

non locating dominating graph then G has a RD -set that

contains the 1 -vertex non locating dominating set of G .

Theorem 3.2: Let G be a 1 -vertex non locating dominating

graph with a 1 -vertex non locating dominating set S . If for

every 1 -vertex non locating dominating set of the graph G

there exist a vertex u V G S such that S u S then

1RD G S otherwise RD G S .

Proof: Let S be the set of all 1 -vertex non locating

dominating set of G . By Theorem 3.1, G has atleast one

RD -set which contains S and hence RD G S .

Suppose if for every S S , there exist some vertex

u V G S such that S u S , then none of the 1 -

vertex non locating dominating set can be the RD -set of G .

Hence -set of S RD G and S u is the RD -set of G

with cardinality 1S .

Suppose S v S for all v V G S then S will be

the RD -set of G and hence RD G S .

Theorem 3.3: Let G be any graph which is not a 1 -vertex

non locating dominating graph. If G is a wholly located

dominated graph then 1RD G RD G otherwise

RD G RD G .

Proof: By comparing Lemma 3.2 with the hypothesis of the

theorem we get 1RD G RD G

Claim: Suppose G is a wholly located dominated graph then

none of the RD -set of G would be the RD -set of G .

Since G is a wholly located dominated graph, with respect to

every RD -set S of the graph G there exist some vertex

v V G S such that S v S and hence .G

N v S

Thus none of the RD -set of G will be the RD -set of G .

Case 1: G is a wholly located dominated graph

Suppose G has a LD -set G

S with cardinality RD G then

GS is the LD -set G or

GS is the 1 -vertex non locating

dominating set of G .

By hypothesis of the theorem G

S cannot be the 1 -vertex non

locating dominating set of G . And if G

S with cardinality

RD G is a RD -set of G then it cannot be the RD -set of

G by claim. Hence G doesn’t have a LD -set with

cardinality RD G and thus 1RD G RD G .

Case 2: G is not a wholly located dominated graph

G has a RD -set S such that S v S for all v V G S

and hence S will be a LD -set of G . But as

1RD G RD G , S is the RD -set of G and hence

RD G RD G .

Remark 3.1: Let G be any graph and S denote the set of all

1 -vertex non locating dominating set of G . Now schematic

representation of procedure of finding location domination

number of complement of a graph is given below.


5 of 36

Acknowledgment

This research is supported by UGC scheme RGNF. Award

letterF1-17.1/2014-15/RGNF-2014-15-SC-TAM-80373/

(SAIII/Website).

References

[1] Ore, O., Theory of Graphs, Amer. Math. Soc. Colloq.

Publ., 38 (Amer. Math. Soc., Providence, RI), 1962.

[2] Harary, F., Graph theory. Addison-Wesley Publishing

Co., Reading, Mass.-Menlo Park, Calif.-London, 1969.

[3] Slater, P., J., 1988, “Dominating and reference sets in a

graph”, J. Math. Phys. Sci., 22, pp. 445-455.

[4] Slater, P., J., 1987, “Dominating and location in acyclic

in graphs, Networks”, 17, pp. 55-64.

[5] C. Hernando, M. Mora and I. M. Pelayo, 2014,

Nordhaus-Gaddum bounds for locating domination, Eur.

J. Combin., 36, pp. 1–6.

[6] Hernando, C., Mora, M., and Pelayo, I., M., Locating

domination in bipartite graphs and their complements.

arXiv:1711.01951v2, 2017.

[7] Hernando, C., Mora, M., and Pelayo, I., M., 2015, “On

global location-domination in graphs”, Ars Math.

Contemp., 8 (2), pp. 365–379.

[8] Rajasekar, G., and Nagarajan, K., “Algorithm for

finding Location Domination Number of Corona

Product of Graphs”, submitted for publication.

[9] Rajasekar, G., and Nagarajan, K., “Location Domination

Number of Sum of Graphs”, submitted for publication.


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