ISRN Discrete MathematicsVolume 2011 (2011), Article ID 681274, 13 pagesdoi:10.5402/2011/681274

Research Article

-Tuple Total Domination in Complementary Prisms

Adel P. Kazemi

Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 5619911367, Ardabil, Iran

Received 29 September 2011; Accepted 30 October 2011

Academic Editor: W. Wang

Copyright © 2011 Adel P. Kazemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a positive integer, and let be a graph with minimum degree at least . In their study (2010), Henning and Kazemi defined the -tuple total domination number of as the minimum cardinality of a -tuple total dominating set of , which is a vertex set such that every vertex of is adjacent to at least vertices in it. If is the complement of , the complementary prism of is the graph formed from the disjoint union of and by adding the edges of a perfect matching between the corresponding vertices of and . In this paper, we extend some of the results of Haynes et al. (2009) for the -tuple total domination number and also obtain some other new results. Also we find the -tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.

1. Introduction

In this paper, is a simple graph with the vertex set and the edge set . The order of is denoted by . The open neighborhood and the closed neighborhood of a vertex are and , respectively. Also the degree of is . Similarly, the open neighborhood and the closed neighborhood of a set are and , respectively. The complement of is the graph with the vertex set and the edge set . The minimum and maximum degree of are denoted by and , respectively. We also write , , and for the complete graph, cycle, and path of order , respectively, while and denote the subgraph induced on by a vertex set , and the complete -partite graph, respectively.

Haynes et al. in [1] have defined complementary product of two graphs that generalizes the Cartesian product of two graphs. Let and be two graphs. For each and , the complementary product is a graph with the vertex set and is an edge in (1)if , , and or if , , and , or(2)if , , and or if , , and .

In other words, for each , we replace by a copy of if is in and by a copy of its complement if is not in , and for each , we replace each by a copy of if and by a copy of if . If (resp., ), we write simply (resp., ). Thus, is the graph obtained by replacing each vertex of by a copy of if and by a copy of if and replacing each vertex of by a copy of . We recall that the Cartesian product of two graphs and is the complementary product . The special complementary product , where , is called the complementary prism of and denoted by . For example, the graph is the Petersen graph. Also, if , the graph is the corona , where the corona of a graph is the graph obtained from by attaching a pendant edge to each vertex of . We notice that .

In [2], Henning and Kazemi introduced the -tuple total domination number of graphs. Let be a positive integer. A subset of is a -tuple total dominating set of , abbreviated kTDS, if for every vertex , , that is, is a kTDS of if every vertex of has at least neighbors in . The -tuple total domination number of is the minimum cardinality of a kTDS of . We remark that a 1-tuple total domination is the well-studied total domination number. Thus, . For a graph to have a -tuple total dominating set, its minimum degree is at least . Since every -tuple total dominating set is also a -tuple total dominating set, we note that for all graphs with minimum degree at least . A kTDS of cardinality is called a -set. When , a 2-tuple total dominating set is called a double total dominating set, abbreviated DTDS, and the 2-tuple total domination number is called the double total domination number. The redundancy involved in -tuple total domination makes it useful in many applications. The paper in [3] gives more information about the -tuple total domination number of a graph.

In [4], Haynes et al. discussed the domination and total domination number of complementary prisms. In this paper, we extend some of their results for the -tuple total domination number and obtain some other results. More exactly, we find some useful lower and upper bounds for the -tuple total domination number of the complementary prism in terms on the order of , , , , and , in which some of the bounds are sharp. Also we find this number for , when is a cycle, a path, or a complete multipartite graph.

Through of this paper, is a positive integer, and for simplicity, we assume that is the disjoint union with and such that . The vertices and are called the corresponding vertices. Also for a subset , we show its corresponding subset in with . The next known results are useful for our investigations.

Proposition A (Haynes et al. [2]). If is a path or a cycle of order such that or is the corona graph , where , then .

Proposition B (Henning and Kazemi [4]). Let be an integer, and let be a complete -partite graph, where . (i)If , then ,(ii)if and , then ,(iii)if and , then .

Proposition C (Henning and Kazemi [5]). Let be a graph of order with . Then

Proposition D (Henning and Kazemi [5]). Let be a graph of order with , and let be a kTDS of . Then for every vertex of degree in , .

2. Some Bounds

The next two theorems state some lower and upper bounds for .

Theorem 2.1. If is a graph of order with , then

Proof. Since for every -set the set is a kTDS of , we get . Similarly, we have . Therefore For proving , let be a kTDS of . Then is a ()TDS of and is a ()TDS of . Since every vertex of (resp., ) is adjacent to only one vertex of (resp., ). Therefore

The given bounds in Theorem 2.1 are sharp. Let be a -regular graph of odd order . Then and are ()- and -regular, respectively, and Proposition D implies and . Therefore The Harary graphs [6] are a family of this kind of graphs. We recall that the Harary graph is a -regular graph with the vertex set and every vertex is adjacent to the vertices in the set

Theorem 2.2. If is a graph of order with , then and the lower bound is sharp for .

Proof. Trivially . Let be a kTDS of , and let be a kTDS of . Then is a kTDS of , and so Proposition A implies that, if , then the lower bound is sharp for all paths and cycles of order , where , and for the corona graph , where .

In special case , we get the following result in [1].

Corollary 2.3 (see [1]). If and have no isolated vertices, then

3. The Complementary Prism of Some Graphs

In this section, we calculate the -tuple total domination number of the complementary prism , when is a complete multipartite graph, a cycle, or a path. First let be a complete -partite graph with the vertex partition such that for each , and . Then , where denotes the corresponding set of . Trivially for to have -tuple total domination number we should have . In the next five propositions, we calculate this number for the complementary prism of the complete -partite graph . First we state the following key lemma which has an easy proof that is left to the reader.

Lemma 3.1. Let be a complete -partite graph with . If is a kTDS of , then for each , . Furthermore, if for some , then .

Proposition 3.2. Let be a complete -partite graph with . Then where , and .

Proof. Let be an arbitrary kTDS of , and let . Proposition D implies that for every , or and . Also if and , it implies . Therefore , and hence . Now we set as a -set such that , for each . Since is a TDS of of cardinality , we get .

Corollary 3.3 (see [1]). If , then .

Proposition 3.4. If is a complete -partite graph with , then

Proof. We discuss .Case 1 (). It follows by and Lemma 3.1 that, for every -tuple total dominating set of , for and for . Then Now we set such that is a ()-subset of , for . Since is a kTDS of of cardinality , we have .Case 2 (). It follows by and Lemma 3.1 that, for every kTDS of , is a subset of and also every vertex of is adjacent to at least vertices of . Thus either for each and or for some . Therefore Now we set such that is a ()-subset of for and is a ()-subset of such that . Since is a kTDS of of cardinality , we get .

Now let be a complete -partite graph with , and let be a minimal kTDS of . Then , by Lemma 3.1. We notice that if , for some , then we may improve and obtain another kTDS of cardinality such that (since every vertex in (respectively ) is adjacent to only one vertex in (respectively )). Therefore, we may assume that for every minimal kTDS of , we have .

Now let be a minimal kTDS of , and let be a set of cardinality . We consider the following two cases.

Case 1 (). In this case, if , we have such that if and only if , and otherwise. If , then and if and , then we have also . Hence Comparing (3.6), (3.7) shows that for if is a set of vertices such that and for and for , then is a minimum kTDS of and

Case 2 (). In this case, for each we have . We continue our discussion in the next subcases.Subcase 1 ( or ). Then obviously . If for we consider , then is a minimum kTDS of and Subcase 2 (). Then obviously . If we set such that , and when , then is a minimum kTDS of and Subcase 3 (). Then obviously . If is a -set, then is a minimum kTDS of , and Proposition B implies Subcase 4 (). Then obviously . If is a -set, then is a minimum kTDS of , and Proposition B implies

Now let be a complete -partite graph with , and let is a minimal kTDS of . In this case, we may similarly assume that . Also it can be easily seen that if for some , then equality (3.8) holds. Thus let . Then obviously . If we choose a set such that is a -set and for , then is a minimum kTDS of , and Proposition B implies Comparing (3.9), (3.10), (3.11), (3.12), and (3.13) with (3.8) shows that we have proved the following propositions.

Proposition 3.5. Let be a complete -partite graph with . Then .

Proposition 3.6. Let be a complete -partite graph with . Then

Proposition 3.7. Let be a complete -partite graph with . Then

We now determine the -tuple total domination number of the complementary prism , where . Here we assume that , , and . Proposition D implies that . In many references, for example, in [1], it can be seen that, for , and trivially we can prove Hence Theorem 2.1 implies that where , and also Theorem 2.2 implies that where . In chain (3.19) we need to calculate , which is done by the next proposition.

Proposition 3.8. If is a cycle of order , then

Proof. Proposition C implies that . If , then, for each , the set is a DTDS of and so . If , then it can be easily verified that . Now since and are double total dominating sets of , where and , respectively, we get . Finally if , then is 2-regular and Proposition D implies .

Proposition 3.9. If , then .

Proof. Let . equalities (3.18), (3.19) and Propositions C and 3.8 imply If , then , and so . Thus we assume . Then and hence . Now let be a -set. If , then , for some two adjacent vertices , and so . Thus we assume . Without loss of generality, let . Since , we continue our proof in the following two cases.Case 1 (). Then . We note that, for every , . This implies , and since must be dominated by , we have . If , then and so . Let . If , again . But implies . Let . The condition implies . Therefore for at least one vertex , and hence .Case 2 ( (similarly )). Case 1 implies . Then . Again we see that, for every , and so .Therefore, in the previous all cases, we proved that and chain (3.21) implies .

Corollary 3.10. If , then

Now we determine the exact amount of for . Obviously . In the next proposition we calculate it when .

Proposition 3.11. Let . Then

Proof. Theorem 2.2 with equalities (3.16) and (3.17) implies and if and , then and if and , then If and , then the sets and are total dominating sets of , respectively. Hence chain (3.24) implies for . Now we assume . For , since the sets and are two total dominating sets of of cardinality , where and , respectively, we have , by chain (3.25). Now let . We assume that is a TDS of . Obviously . If and , then , and hence , where . This implies Now let . If , then and if , then It can be calculated that Then by chains (3.25) and (3.26) we have If , then the sets and are total dominating sets of of cardinality when and , respectively. Hence , by chain (3.32). If also , the sets and are total dominating sets of of cardinality when and , respectively. Hence , by chain (3.33).

Finally we determine the -tuple total domination number of the complementary prism , where . We recall that , , and . In many references, for example, in [1], it can be seen that, for , and trivially , where . Therefore, by Theorems 2.1 and 2.2, for , we have the following chain: It can be easily proved that , where . Next proposition calculates when .

Proposition 3.12. Let . Then

Proof. Let be a -set of the induced path of . Since is a TDS of , we have Let . Then chains (3.34), (3.35), (3.37) imply . Since . Now let , and let be a TDS of . Obviously . In all cases, (i) and , (ii) and , and (iii) , then similar to the proof of Proposition 3.11, it can be verified that Hence chain (3.37) completes the proof of our proposition.

Propositions 3.11 and 3.12 imply the next result in [1].

Corollary 3.13 (see [1]). If with order , then

4. Problems

If we look carefully at the propositions of Section 3, we obtain the following result.

Proposition 4.1. (i) Let be a cycle or a path of order . Then if and only if . (ii) Let be a cycle of order or a path of order . Then if and only if . (iii) Let be a cycle of order . Then (iv) Let be a cycle of order . Then

Therefore it is natural that we state the following problem.

Problem 1. Characterize graphs with(1), (2).

References

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