Inverse Secure Restrained Domination in the Join ... - Semantic Scholar

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6676-6679 © Research India Publications. http://www.ripublication.com

Inverse Secure Restrained Domination in the Join and Corona of Graphs Edward M. Kiunisala Professor, Mathematics Department, College of Arts and Sciences, Cebu Normal University, Philippines.

Enrico L. Enriquez Professor, Mathematics Department, School of Arts and Sciences, University of San Carlos, Philippines.

it is desirable to place a few guards as possible. In [5, 6], Enriquez and Canoy, introduced the concepts of secure convex and restrained convex domination in graphs. In [11], Pushpam and Suseendran studied few properties of secure restrained domination number of certain classes of graphs and values for trees, unicyclic graphs, split graphs evaluate and generalized Petersen graphs. The inverse domination in graph was first found in the paper of Kulli [14] and further read in [8, 13]. Moreover, for the general concepts, the reader may refer to [9]. In this study, we introduce a new domination parameter, the inverse secure restrained domination in graphs. The main results of this study is the characterization of the inverse secure restrained dominating sets in the join and corona of two graphs. A graph is a pair , , where is a finite nonempty set called the vertex-set of and is a set of unordered pairs , (or simply ) of distinct elements from called the edge-set of . The elements of are called vertices and the cardinality | | of is the order of . The elements of are called edges and the cardinality | | of is the size of . If | | 1, then is called a trivial graph. If , then is called an empty graph. The open neighborhood of a vertex . The is the set are called neighbors of . The closed elements of neighborhood of is the set . If , the open neighborhood of in is the . The closed neighborhood of X in G set . When no is the set . will be denoted by N[x] confusion arises, [resp. N(x)]. The join of two graphs and is the graph with vertex-set and edgeset , . A nonempty subset S of , where is any graph, is a clique in if the graph induced by is complete. A subset of is a dominating set of if for every \ , there exists such that , i.e., . The domination number of is the smallest cardinality of a dominating set of . A dominating set of is a secure dominating set of if for each \ , there exists such that and the set \ is a dominating set of . The minimum cardinality of a secure dominating set of , denoted , is called the secure domination number of . A by is called a -set of secure dominating set of cardinality . A set is a restrained dominating set if every vertex not in is adjacent to a vertex in and to a vertex in \ . Alternately, a subset of is a restrained

AbstractLet be a connected simple graph. A restrained dominating set of the vertex set of , , is a secure restrained dominating set of if for each \ , there exists such that and the set \ is a restrained dominating set of . The minimum cardinality of a , is secure restrained dominating set of , denoted by called the secure restrained domination number of . A is called secure restrained dominating set of cardinality -set of . A secure restrained dominating set of the a vertex set of is an inverse secure restrained dominating set if \ where is a minimum secure restrained dominating set of . The minimum cardinality of an inverse , is secure restrained dominating set of , denoted by called an inverse secure restrained domination number of . is A secure restrained dominating set of cardinality of . In this paper, we characterize the called a inverse secure restrained dominating sets in the join and corona of two graphs and give some important results. Mathematics Subject Classification: 05C69 Keywords: dominating set, secure dominating set, secure restrained dominating set, inverse secure restrained dominating set.

INTRODUCTION Domination in graph was introduced by Claude Berge in 1958 and Oystein Ore in 1962 [12]. However, it was not until 1977, following an article [3] by Ernie Cockayne and Stephen Hedetniemi, that domination in graphs became an area of study by many researchers. One type of domination parameter is the secure domination in graphs. This was studied and introduced by E.J. Cockayne et.al [1, 4]. Secure dominating sets can be applied as protection strategies by minimizing the number of guards to secure a system so as to be cost effective as possible. Other type of domination parameter is the restrained domination number in a graph. This was introduced by Telle and Proskurowski [10] indirectly as a vertex partitioning problem. One practical application of restrained domination is that of prisoners and guards. Here, each vertex not in the restrained dominating set corresponds to a position of a prisoner, and every vertex in the restrained dominating set corresponds to a position of a guard. To effect security, each prisoner’s position is observed by a guard’s position. To protect the rights of prisoners, each prisoner’s position is seen by at least one other prisoner’s position. To be cost effective,

6676

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6676-6679 © Research India Publications. http://www.ripublication.com dominating set if and \ is a subgraph without isolated vertices. A restrained dominating set of is a secure restrained dominating set of if for each \ , there exists such that and the set \ is a restrained dominating set of . The minimum cardinality of a secure restrained dominating , is called the secure restrained set of , denoted by domination number of . A secure restrained dominating set -set of . Let be a is called a of cardinality minimum dominating set in . The dominating set \ is called an inverse dominating set with respect to . The minimum cardinality of inverse dominating set is called an inverse domination number of and is denoted by . An inverse dominating set of cardinality is -set of . Motivated by the definition of inverse called domination in graph, we define a new domination parameter. Let be a minimum secure restrained dominating set in . The secure restrained dominating set \ is called an inverse secure restrained dominating set with respect to . The minimum cardinality of inverse secure restrained dominating set is called an inverse secure restrained . An domination number of and is denoted by inverse secure restrained dominating set of cardinality -set of G. is called

Theorem 2.5 [7] Let and be connected non-complete graphs. Then a proper subset of is a secure restrained dominating set in if and only if one of the following statements holds: (i) is a secure dominating set of and | | 2. (ii) is a secure dominating set of and | | 2. where and (iii) and is a dominating set of and is a dominating (a) set of ; or is dominating set of and \ \ (b) is a clique in ; or is dominating set of and \ \ (c) is a clique in ; or \ is a clique in and \ (d) \ \ is a clique in . where | | 2 and (iv) and \ \ is a clique in . where and (v) 2 and \ \ is a | | clique in . where | | 2 and (vi) | | 2 . The following result characterized the inverse secure restrained dominating sets in the join of two graphs.

RESULTS does not always exists in a connected nontrivial Since be a family of all graphs with graph G, we denote by inverse secure restrained dominating set. Thus, for the purpose of this study, it is assumed that all connected nontrivial graphs considered (including and ) . belong to the family

Theorem 2.6 Let and be connected non-complete graphs. Then a proper subset of , where \ , is an inverse secure restrained dominating set in if and only if one of the following statements holds: (i) is a secure dominating set of and | | | | 2. (ii) is a secure dominating set of and | | | | 2. where and (iii) and is a dominating set of and is a dominating (a) set of ; or is dominating set of and \ is \ (b) a clique in ; or is dominating set of and \ \ is (c) a clique in ; or is a clique in and \ (d) \ \ is a clique in . \ where | | 2 and (iv) and \ \ is a clique in . where and (v) 2 and \ is a | | \ clique in . where | | 2 and (vi) | | 2 .

and is a secure restrained Remark 2.1 If dominating set of , then there exists \ such that is a secure restrained dominating set of . Theorem 2.2 [2] Let be a graph of order . 1 if and only if

1. Then

Remark 2.3 If is a secure restrained dominating set of a graph , then is a secure dominating set of . Theorem 2.4 Let be a graph of order 1 if and only if .

3. Then

1. Let be a -set of Proof : Suppose that . Then is a secure dominating set of by Remark 2.3. 1, that is, by Theorem 2.2. Thus, . Then 1 by For the converse, suppose that Theorem 2.2. Let be a -set of G. Since and for all \ and for all \ ) (since 3), it follows that is for all also a restrained dominating set of . Since \ is also a restrained dominating set of , it follows that is a secure restrained dominating set of . 1. Accordingly,

Proof: Suppose that is an inverse secure restrained dominating set of . Then is a secure restrained dominating set of . In view of Theorem 2.5(i), S is a secure dominating set of and | | 2. Further, there exists

6677

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6676-6679 © Research India Publications. http://www.ripublication.com \ such that is a minimum secure restrained dominating set of , that is | | | |. If | | 1 then is a complete graph by Theorem 2.4 contrary to our assumption. Thus, | | 2. Thus, is a secure dominating set of and | | | | 2, proving statement (i). Similarly, in view of Theorem 2.5(ii) statement (ii) holds. Now, is a secure restrained dominating set of . In view of Theorem where and 2.5(iiia), is a dominating set of and is a and dominating set of . This proves, statement (iiia). Similarly, any of the statements (iiib), (iiic), (iiid), (iv), (v), or (vi) holds in veiw of (iiib), (iiic), (iiid), (iv), (v), or (vi) respectively of Theorem 2.5. For the converse, suppose that statement (i), (ii), (iii), (iv), (v), or (vi) holds. Consider first that statement (i) holds. is a secure dominating set of and | | | | 2. Then is a secure restrained dominating set of by Theorem 2.5(i). This implies that there exists \ such that is a secure restrained dominating set of by Remark 2.1. \ is Since | | | | and , it follows that an inverse secure restrained dominating set of . Similarly, if statement (ii) holds, then is an inverse secure restrained dominating set of . Next, suppose that statement (iiia) holds. Then by Theorem 2.5(iiia), , is a secure restrained dominating set of . This implies that there exists \ such that is a secure restrained dominating set of by Remark 2.1. Since 1. Thus, | | 2 is a is non-complete, minimum secure restrained dominating set of . This implies that | | | | and . Then \ is an inverse secure restrained dominating set of . Similarly, is an inverse secure restrained dominating set of if any of the statements (iiib), (iiic), (iiid), (iv), (v), or (vi) holds.

2 (by using similar arguments . Thus, 2, then above). Suppose that | | 5. If 2 (by using similar arguments above). If 2, then let where and . Then is a dominating set of . Let \ . If \ , then there exists , say , such that and \ for all \ (and hence \ . If \ , then and for all \ . Thus, is a restrained dominating set of . \ is a dominating set for all Now, \ . Since is connected non-complete graph, and for all , \ is a restrained (and hence , \ ). Thus, dominating set of . Accordingly, is a secure restrained dominating set of . Similarly, if 2, then is a secure restrained dominating set of . In view of Remark 2.1 there exists \ such that is a secure restrained dominating set of . If is a minimum secure | | restrained dominating set of , then 1 by Theorem 3. Since is non-complete, 2. Let , be a 2.4. Suppose that minimum secure restrained dominating set of . If , then is a minimum secure restrained dominating set in contrary to our assumption earlier that , is 2. not a secure dominating set of . Thus, 2. Now, if Similarly, if , Then Thus, , where and . Then , is not a dominating set for some \ (and hence \ ), contrary to our assumption that is 2, that is, a dominating set of . Thus, | | 2. Since | | | | 3, it follows that | | 3. Thus, is the minimum inverse secure restrained dominating set of 3. Finally, suppose that . Accordingly, 3 and 3. Then it can be easily verified that 4. This complete the proofs.

The following result is a quick consequence of Theorem 2.4. Corollary 2.7 Let graphs and let 2, 3, 4,

and and

be connected non-complete . Then 2

1 2, |

5|

2, |

We are needing the following theorem for our next result.

2 5|

Theorem 2.8 [7] Let and be nontrivial connected graphs. A nonempty subset of is a secure restrained dominating set of if and only if for each , is a subgraph without isolated vertices and one \ of the following is satisfied. (i) and is complete. , where is a (ii) for each dominating set of \ . , where is a secure dominating (iii) . set of

.

Proof: Suppose that 1 . Let , where is a dominating set of and is a dominating set of . Then \ is an inverse secure restrained dominating set of by Theorem 2.6(iiia). Thus, | | 2. Suppose that 1. Then is a complete graph and hence and are complete. 2. This is contrary to our assumption. Thus, 2. Let , be a minimum Now, suppose that secure dominating set of . Then \ is an inverse secure restrained dominating set of by | | 2. Since and Theorem 2.6(i). Thus, are non-complete graphs, it follows that is non1 by Theorem 2.4. Thus, complete and hence | | 2. Similarly, if 2, then 2. Next, suppose that 2. Let , be a dominating set of . Since is connected noncomplete graph, | | 3. Suppose that | | 3 or | | 4. Then is a minimum secure dominating set of

The following result characterize the inverse secure restrained dominating sets in the corona of two connected graphs. Theorem 2.9 Let and be nontrivial connected graphs. A nonempty subset of is an inverse secure restrained dominating set of if and only if for each and one of the following is satisfied. , (i) , and (a) is complete, or

6678

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 9 (2016) pp 6676-6679 © Research India Publications. http://www.ripublication.com (b) (ii) (a) (b)

\

is a secure dominating set of . , and is a secure restrained dominating set of , or is a dominating set of for all . \

Corollary 2.10 Let 2. Then

be connected graph and | |.

with

Corollary 2.11 Let and be nontrivial connected graphs. for each , then If is a secure dominating set of ∑ | |.

Proof: Suppose that a nonempty subset of is a an inverse secure restrained dominating set of . Then is a secure restrained dominating set of . Thus, any of the statements (ib) or (iib) is satisfied by Theorem 2.8. Let . Suppose that for each , is non-complete. Then for each , there exists \ is not a dominating set of \ such that . This contradict to our assumption that is a secure restrained dominating set of . Thus, for each , is complete. This proves statement (ia). Let . Suppose that is not a secure dominating set . Then is not a secure dominating set of of is not a secure and hence dominating set of contrary to our assumption that is a must be a secure restrained dominating set of . Thus, secure restrained dominating set of . This proves statement (iia). For the converse, suppose that for each , and one of the following statements (i) or (ii) is for satisfied. Suppose first that (ia) is satisfied. Let each . Since is complete, is a dominating set of for each and hence , is a dominating set of . Let \ . Then for some . If then \ (and hence \ ). If for each \ , then for each (and hence \ ). Thus, is a restrained dominating set \ , then ′ is of . Now, ′ \ . If is complete for each a dominating set of since . If , then ′ is again a dominating set of . Let \ ′. Then since dominate \ ′, then there for some ′. If \ ′ (and hence \ ′) such that exists is complete for each . If since \ ′ \ ′, then for each . Thus, ′ (and hence \ ′) since dominate is a restrained dominating set of . Accordingly, is a secure restrained dominating set of . In view of Remark 2.1 there exists \ such that is a secure \ for restrained dominating set of . Let each . Since is complete, let . This implies that | | | |. Thus, \ is an inverse secure dominating set of . Next suppose that (ib) is satisfied. Then is a secure restrained dominating set of by Theorem 2.8. In view of Remark 2.1 there exists \ such that is a secure restrained dominating is a secure dominating set of , it is set of . Since clear that is a minimum secure restrained dominating set of . Let . Then | | | | | |. Thus, \ is an inverse restrained dominating set of . Similarly, if any of (iia) or (iib) is satisfied, then S is an inverse restrained dominating set of .

ACKNOWLEDGEMENTS The author is truly grateful to the University of San Carlos and Cebu Normal University for giving us moral and even material support in conducting research in the field of Mathematics. Hence the paper.

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9] [10]

[11]

[12] [13]

[14]

6679

C.M. Mynhardt, H.C. Stwart and E. Ungerer Excellent trees and secure domination, Utilitas Math 67(2005) 255-267. E.C. Castillano, R.L. Ugbinada, and S.R. Canoy,Jr. Secure domination in the join of graphs, Applied Mathematical Sciences, 8 (105), 5203-5211 http://dx.doi.org/10.12988/ams.2014.47519 E.J. Cockayne, and S.T. Hedetniemi Towards a theory of domination in graphs, Networks, (1977) 247-261. E.J. Cockayne, O. Favaron and C.M. Mynhardt Secure domination, weak Roman domination and forbidden subgraphs, Bull. Inst. Combin. Appl. 39(2003) 87-100. E.L. Enriquez, and S.R. Canoy,Jr., On a Variant of Convex Domination in a Graph. International Journal of Mathematical Analysis, Vol. 9, 2015, no.32, 15851592. E.L. Enriquez, and S.R. Canoy,Jr., Secure Convex Domination in a Graph. International Journal of Mathematical Analysis, Vol. 9, 2015, no. 7, 317-325. E.L. Enriquez, Secure Restrained Domination in the Join and Corona of Graphs. Global Journal of Pure and Applied Mathematics, Vol. 12, 2016, no. 1, E.M. Kiunisala and F.P. Jamil, Inverse domination Numbers and disjoint domination numbers of graphs under some binary operations, Applied Mathematical Sciences, Vol. 8, 2014, no. 107, 5303-5315. G. Chartrand and P. Zhang, A First Course in Graph Theory. Dover Publication, Inc., New York, 2012. J.A. Telle, A. Proskurowski, Algorithms for Vertex Partitioning Problems on Partial-k Trees, SIAM J. Discrete Mathematics, 10(1997), 529-550. P.R. Pushpam, and C. Suseendran, Secure Restrained Domination in Graphs. Math.Comput.Sci, (2015)9:239-247. DOI 10.1007/s1786-015-0230-4. O. Ore. Theory of Graphs. American Mathematical Society, Provedence, R.I., 1962. T. Tamizh Chelvan, T. Asir and G.S. Grace Prema, Inverse domination in graphs, Lambert Academic Publishing, 2013. V.R. Kulli and S.C. Sigarkanti, Inverse domination in graphs, Nat. Acad. Sci. Letters, 14(1991) 473-475.