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ISRN Discrete Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 681274, 13 pages
http://dx.doi.org/10.5402/2011/681274
Research Article

π‘˜-Tuple Total Domination in Complementary Prisms

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

Received 29 September 2011; Accepted 30 October 2011

Academic Editor: W.Β Wang

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

1. Introduction

In this paper, 𝐺=(𝑉,𝐸) is a simple graph with the vertex set 𝑉 and the edge set 𝐸. The order |𝑉| of 𝐺 is denoted by 𝑛=𝑛(𝐺). The open neighborhood and the closed neighborhood of a vertex π‘£βˆˆπ‘‰ are 𝑁𝐺(𝑣)={π‘’βˆˆπ‘‰(𝐺)βˆ£π‘’π‘£βˆˆπΈ(𝐺)} and 𝑁𝐺[𝑣]=𝑁𝐺(𝑣)βˆͺ{𝑣}, respectively. Also the degree of 𝑣 is deg𝐺(𝑣)=|𝑁𝐺(𝑣)|. Similarly, the open neighborhood and the closed neighborhood of a set π‘†βŠ†π‘‰ are 𝑁𝐺⋃(𝑆)=π‘£βˆˆπ‘†π‘(𝑣) and 𝑁𝐺[𝑆]=𝑁𝐺(𝑆)βˆͺ𝑆, respectively. The complement of 𝐺 is the graph 𝐺 with the vertex set 𝑉(𝐺)=𝑉(𝐺) and the edge set 𝐸(𝐺)={π‘’π‘£βˆ£π‘’π‘£βˆ‰πΈ(𝐺)}. The minimum and maximum degree of 𝐺 are denoted by 𝛿=𝛿(𝐺) and Ξ”=Ξ”(𝐺), respectively. We also write 𝐾𝑛, 𝐢𝑛, and 𝑃𝑛 for the complete graph, cycle, and path of order 𝑛, respectively, while 𝐺[𝑆] and 𝐾𝑛1,𝑛2,…,𝑛𝑝 denote the subgraph induced on 𝐺 by a vertex set 𝑆, and the complete 𝑝-partite graph, respectively.

Haynes et al. in [1] have defined complementary product of two graphs that generalizes the Cartesian product of two graphs. Let 𝐺 and 𝐻 be two graphs. For each π‘…βŠ†π‘‰(𝐺) and π‘†βŠ†π‘‰(𝐻), the complementary product 𝐺(𝑅)░𝐻(𝑆) is a graph with the vertex set {(𝑒𝑖,𝑣𝑗)βˆ£π‘’π‘–βˆˆπ‘‰(𝐺),π‘£π‘–βˆˆπ‘‰(𝐻)} and (𝑒𝑖,𝑣𝑗)(π‘’β„Ž,π‘£π‘˜) is an edge in 𝐸(𝐺(𝑅)░𝐻(𝑆))(1)if 𝑖=β„Ž, π‘’π‘–βˆˆπ‘…, and π‘£π‘—π‘£π‘˜βˆˆπΈ(𝐻) or if 𝑖=β„Ž, π‘’π‘–βˆ‰π‘…, and π‘£π‘—π‘£π‘˜βˆ‰πΈ(𝐻), or(2)if 𝑗=π‘˜, π‘£π‘—βˆˆπ‘†, and π‘’π‘–π‘’β„ŽβˆˆπΈ(𝐺) or if 𝑗=π‘˜, π‘£π‘—βˆ‰π‘†, and π‘’π‘–π‘’β„Žβˆ‰πΈ(𝐺).

In other words, for each π‘’π‘–βˆˆπ‘‰(𝐺), we replace 𝑒𝑖 by a copy of 𝐻 if 𝑒𝑖 is in 𝑅 and by a copy of its complement 𝐻 if 𝑒𝑖 is not in 𝑅, and for each π‘£π‘—βˆˆπ‘‰(𝐻), we replace each 𝑣𝑗 by a copy of 𝐺 if π‘£π‘—βˆˆπ‘† and by a copy of 𝐺 if π‘£π‘—βˆ‰π‘†. If 𝑅=𝑉(𝐺) (resp., 𝑆=𝑉(𝐻)), we write simply 𝐺░𝐻(𝑆) (resp., 𝐺(𝑅)░𝐻). Thus, 𝐺░𝐻(𝑆) is the graph obtained by replacing each vertex 𝑣 of 𝐻 by a copy of 𝐺 if π‘£βˆˆπ‘† and by a copy of 𝐺 if π‘£βˆ‰π‘† and replacing each vertex 𝑒 of 𝐺 by a copy of 𝐻. We recall that the Cartesian product 𝐺░𝐻 of two graphs 𝐺 and 𝐻 is the complementary product 𝐺(𝑉(𝐺))░𝐻(𝑉(𝐻)). The special complementary product 𝐺░𝐾2(𝑆), where |𝑆|=1, is called the complementary prism of 𝐺 and denoted by 𝐺𝐺. For example, the graph 𝐢5𝐢5 is the Petersen graph. Also, if 𝐺=𝐾𝑛, the graph 𝐾𝑛𝐾𝑛 is the corona πΎπ‘›βˆ˜πΎ1, where the corona 𝐺∘𝐾1 of a graph 𝐺 is the graph obtained from 𝐺 by attaching a pendant edge to each vertex of 𝐺. We notice that 𝛿(𝐺𝐺)=min{𝛿(𝐺),𝛿(𝐺)}+1.

In [2], Henning and Kazemi introduced the π‘˜-tuple total domination number of graphs. Let π‘˜ be a positive integer. A subset 𝑆 of 𝑉 is a π‘˜-tuple total dominating set of 𝐺, abbreviated kTDS, if for every vertex π‘£βˆˆπ‘‰, |𝑁(𝑣)βˆ©π‘†|β‰₯π‘˜, that is, 𝑆 is a kTDS of 𝐺 if every vertex of 𝑉 has at least π‘˜ neighbors in 𝑆. The π‘˜-tuple total domination number π›ΎΓ—π‘˜,𝑑(𝐺) of 𝐺 is the minimum cardinality of a kTDS of 𝐺. We remark that a 1-tuple total domination is the well-studied total domination number. Thus, 𝛾𝑑(𝐺)=𝛾×1,𝑑(𝐺). For a graph to have a π‘˜-tuple total dominating set, its minimum degree is at least π‘˜. Since every (π‘˜+1)-tuple total dominating set is also a π‘˜-tuple total dominating set, we note that π›ΎΓ—π‘˜,𝑑(𝐺)≀𝛾×(π‘˜+1),𝑑(𝐺) for all graphs with minimum degree at least π‘˜+1. A kTDS of cardinality π›ΎΓ—π‘˜,𝑑(𝐺) is called a π›ΎΓ—π‘˜,𝑑(𝐺)-set. When π‘˜=2, 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 π‘˜-tuple total domination makes it useful in many applications. The paper in [3] gives more information about the π‘˜-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 π‘˜-tuple total domination number and obtain some other results. More exactly, we find some useful lower and upper bounds for the π‘˜-tuple total domination number of the complementary prism 𝐺𝐺 in terms on the order of 𝐺, π›ΎΓ—π‘˜,𝑑(𝐺), π›ΎΓ—π‘˜,𝑑(𝐺), 𝛾×(π‘˜βˆ’1),𝑑(𝐺), and 𝛾×(π‘˜βˆ’1),𝑑(𝐺), in which some of the bounds are sharp. Also we find this number for 𝐺𝐺, when 𝐺 is a cycle, a path, or a complete multipartite graph.

Through of this paper, π‘˜ is a positive integer, and for simplicity, we assume that 𝑉(𝐺𝐺) is the disjoint union 𝑉(𝐺)βˆͺ𝑉(𝐺) with 𝑉(𝐺)={π‘£βˆ£π‘£βˆˆπ‘‰(𝐺)} and 𝐸(𝐺𝐺)=𝐸(𝐺)βˆͺ𝐸(𝐺)βˆͺ{π‘£π‘£βˆ£π‘£βˆˆπ‘‰(𝐺)} such that 𝐸(𝐺)={π‘’π‘£βˆ£π‘’π‘£βˆ‰πΈ(𝐺)}. The vertices 𝑣 and 𝑣 are called the corresponding vertices. Also for a subset π‘‹βŠ†π‘‰(𝐺), we show its corresponding subset in 𝐺 with 𝑋. The next known results are useful for our investigations.

Proposition A (Haynes et al. [2]). If 𝐺 is a path or a cycle of order 𝑛β‰₯5 such that 𝑛≑2(mod4) or is the corona graph πΎπ‘›βˆ˜πΎ1, where 𝑛β‰₯3, then 𝛾𝑑(𝐺𝐺)=𝛾𝑑(𝐺).

Proposition B (Henning and Kazemi [4]). Let 𝑝β‰₯2 be an integer, and let 𝐺=𝐾𝑛1,𝑛2,…,𝑛𝑝 be a complete 𝑝-partite graph, where 𝑛1≀𝑛2≀⋯≀𝑛𝑝.(i)If π‘˜<𝑝, then π›ΎΓ—π‘˜,𝑑(𝐺)=π‘˜+1,(ii)if π‘˜=𝑝 and βˆ‘π‘˜βˆ’1𝑖=1𝑛𝑖β‰₯π‘˜, then π›ΎΓ—π‘˜,𝑑(𝐺)=π‘˜+2,(iii)if 2≀𝑝<π‘˜ and βŒˆπ‘˜/(π‘βˆ’1)βŒ‰β‰€π‘›1≀𝑛2≀⋯≀𝑛𝑝, then π›ΎΓ—π‘˜,𝑑(𝐺)=βŒˆπ‘˜π‘/(π‘βˆ’1)βŒ‰.

Proposition C (Henning and Kazemi [5]). Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺)β‰₯π‘˜. Then π›ΎΓ—π‘˜,𝑑(𝐺)β‰₯maxπ‘˜+1,π‘˜π‘›.Ξ”(𝐺)ξƒ’ξ‚Ό(1.1)

Proposition D (Henning and Kazemi [5]). Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺)β‰₯π‘˜, and let 𝑆 be a kTDS of 𝐺. Then for every vertex 𝑣 of degree π‘˜ in 𝐺, 𝑁𝐺(𝑣)βŠ†π‘†.

2. Some Bounds

The next two theorems state some lower and upper bounds for π›ΎΓ—π‘˜,𝑑(𝐺𝐺).

Theorem 2.1. If 𝐺 is a graph of order 𝑛 with 2β‰€π‘˜β‰€min{𝛿(𝐺),𝛿(𝐺)}, then 𝛾×(π‘˜βˆ’1),𝑑(𝐺)+𝛾×(π‘˜βˆ’1),π‘‘ξ‚€πΊξ‚β‰€π›ΎΓ—π‘˜,𝑑𝐺𝐺𝛾≀minΓ—(π‘˜βˆ’1),𝑑(𝐺),𝛾×(π‘˜βˆ’1),𝑑𝐺+𝑛.(2.1)

Proof. Since for every 𝛾×(π‘˜βˆ’1),𝑑(𝐺)-set 𝐷 the set 𝐷βˆͺ𝑉(𝐺) is a kTDS of 𝐺𝐺, we get π›ΎΓ—π‘˜,𝑑(𝐺𝐺)≀𝛾×(π‘˜βˆ’1),𝑑(𝐺)+𝑛. Similarly, we have π›ΎΓ—π‘˜,𝑑(𝐺𝐺)≀𝛾×(π‘˜βˆ’1),𝑑(𝐺)+𝑛. Therefore π›ΎΓ—π‘˜,𝑑𝐺𝐺𝛾≀minΓ—(π‘˜βˆ’1),𝑑(𝐺),𝛾×(π‘˜βˆ’1),𝑑𝐺+𝑛.(2.2)
For proving 𝛾×(π‘˜βˆ’1),𝑑(𝐺)+𝛾×(π‘˜βˆ’1),𝑑(𝐺)β‰€π›ΎΓ—π‘˜,𝑑(𝐺𝐺), let 𝐷 be a kTDS of 𝐺𝐺. Then π·βˆ©π‘‰(𝐺) is a (π‘˜βˆ’1)TDS of 𝐺 and π·βˆ©π‘‰(𝐺) is a (π‘˜βˆ’1)TDS of 𝐺. Since every vertex of 𝑉(𝐺) (resp., 𝑉(𝐺)) is adjacent to only one vertex of 𝑉(𝐺) (resp., 𝑉(𝐺)). Therefore 𝛾×(π‘˜βˆ’1),𝑑(𝐺)+𝛾×(π‘˜βˆ’1),𝑑𝐺≀||||+|||ξ‚€π·βˆ©π‘‰(𝐺)π·βˆ©π‘‰πΊξ‚|||=||𝐷||=π›ΎΓ—π‘˜,𝑑𝐺𝐺.(2.3)

The given bounds in Theorem 2.1 are sharp. Let 𝐺 be a (π‘˜βˆ’1)-regular graph of odd order 𝑛=2π‘˜βˆ’1. Then 𝐺 and 𝐺𝐺 are (π‘˜βˆ’1)- and π‘˜-regular, respectively, and Proposition D implies π›ΎΓ—π‘˜,𝑑(𝐺𝐺)=2𝑛 and 𝛾×(π‘˜βˆ’1),𝑑(𝐺)=𝛾×(π‘˜βˆ’1),𝑑(𝐺)=𝑛. Therefore𝛾×(π‘˜βˆ’1),𝑑(𝐺)+𝛾×(π‘˜βˆ’1),𝑑𝐺=π›ΎΓ—π‘˜,𝑑𝐺𝐺𝛾=minΓ—(π‘˜βˆ’1),𝑑(𝐺),𝛾×(π‘˜βˆ’1),𝑑𝐺+𝑛.(2.4) The Harary graphs 𝐻2π‘š,4π‘š+1 [6] are a family of this kind of graphs. We recall that the Harary graph 𝐻2π‘š,𝑛 is a 2π‘š-regular graph with the vertex set {π‘–βˆ£1≀𝑖≀𝑛} and every vertex 𝑖 is adjacent to the 2π‘š vertices in the setξ‚†πœŽπ‘–π‘—βˆ£πœŽπ‘–π‘—β‰‘π‘–+𝑗(mod𝑛)orπœŽπ‘–π‘—β‰‘π‘–βˆ’π‘—(mod𝑛),for.1β‰€π‘—β‰€π‘š(2.5)

Theorem 2.2. If 𝐺 is a graph of order 𝑛 with 1β‰€π‘˜β‰€min{𝛿(𝐺),𝛿(𝐺)}, then 𝛾maxΓ—π‘˜,𝑑(𝐺),π›ΎΓ—π‘˜,π‘‘ξ‚€πΊξ‚ξ‚‡β‰€π›ΎΓ—π‘˜,π‘‘ξ‚€πΊπΊξ‚β‰€π›ΎΓ—π‘˜,𝑑(𝐺)+π›ΎΓ—π‘˜,𝑑𝐺,(2.6) and the lower bound is sharp for π‘˜=1.

Proof. Trivially max{π›ΎΓ—π‘˜,𝑑(𝐺),π›ΎΓ—π‘˜,𝑑(𝐺)}β‰€π›ΎΓ—π‘˜,𝑑(𝐺𝐺). Let 𝑆 be a kTDS of 𝐺, and let π‘†ξ…ž be a kTDS of 𝐺. Then 𝑆βˆͺπ‘†ξ…ž is a kTDS of 𝐺𝐺, and so π›ΎΓ—π‘˜,π‘‘ξ‚€πΊπΊξ‚β‰€π›ΎΓ—π‘˜,𝑑(𝐺)+π›ΎΓ—π‘˜,𝑑𝐺.(2.7) Proposition A implies that, if π‘˜=1, then the lower bound is sharp for all paths and cycles of order 𝑛β‰₯5, where 𝑛≑2(mod4), and for the corona graph πΎπ‘›βˆ˜πΎ1, where 𝑛β‰₯3.

In special case π‘˜=1, we get the following result in [1].

Corollary 2.3 (see [1]). If 𝐺 and 𝐺 have no isolated vertices, then 𝛾max𝑑(𝐺),𝛾𝑑𝐺≀𝛾𝑑𝐺𝐺≀𝛾𝑑(𝐺)+𝛾𝑑𝐺.(2.8)

3. The Complementary Prism of Some Graphs

In this section, we calculate the π‘˜-tuple total domination number of the complementary prism 𝐺𝐺, when 𝐺 is a complete multipartite graph, a cycle, or a path. First let 𝐺=𝐾𝑛1,𝑛2,…,𝑛𝑝 be a complete 𝑝-partite graph with the vertex partition 𝑉(𝐺)=𝑋1βˆͺ𝑋2βˆͺβ‹―βˆͺ𝑋𝑝 such that for each 1≀𝑖≀𝑝, |𝑋𝑖|=𝑛𝑖 and 𝑛1≀𝑛2≀⋯≀𝑛𝑝. Then 𝑉(𝐺⋃𝐺)=1≀𝑖≀𝑝(𝑋𝑖βˆͺ𝑋𝑖), where 𝑋𝑖 denotes the corresponding set of 𝑋𝑖. Trivially for 𝐺𝐺 to have π‘˜-tuple total domination number we should have π‘˜β‰€π‘›1≀𝑛2≀⋯≀𝑛𝑝. In the next five propositions, we calculate this number for the complementary prism of the complete 𝑝-partite graph 𝐺. First we state the following key lemma which has an easy proof that is left to the reader.

Lemma 3.1. Let 𝐺=𝐾𝑛1,𝑛2,…,𝑛𝑝 be a complete 𝑝-partite graph with 𝑉(𝐺⋃𝐺)=1≀𝑖≀𝑝(𝑋𝑖βˆͺ𝑋𝑖). If 𝑆 is a kTDS of 𝐺𝐺, then for each 1≀𝑖≀𝑝, |π‘†βˆ©π‘‹π‘–|β‰₯π‘˜. Furthermore, if |π‘†βˆ©π‘‹π‘–|=π‘˜ for some 𝑖, then |π‘†βˆ©π‘‹π‘–|β‰₯π‘˜.

Proposition 3.2. Let 𝐺=𝐾𝑛1,𝑛2,…,𝑛𝑝 be a complete 𝑝-partite graph with 1≀𝑛1≀𝑛2≀⋯≀𝑛𝑝. Then 𝛾𝑑𝐺𝐺=2π‘βˆ’π›Ό,(3.1) where 𝛼=|{π‘–βˆ£1≀𝑖≀𝑝, and 𝑛𝑖=1}|.

Proof. Let 𝑆 be an arbitrary kTDS of 𝐺𝐺, and let 𝑛1=𝑛2=β‹―=𝑛𝛼=1<𝑛𝛼+1≀⋯≀𝑛𝑝. Proposition D implies that for every 1≀𝑖≀𝑝, |π‘†βˆ©π‘‹π‘–|β‰₯2 or |π‘†βˆ©π‘‹π‘–|=1 and |π‘†βˆ©π‘‹π‘–|β‰₯1. Also if |𝑋𝑖|=1 and |π‘†βˆ©π‘‹π‘–|=0, it implies |π‘†βˆ©π‘‹π‘–|=1. Therefore |𝑆|β‰₯𝛼+2(π‘βˆ’π›Ό)=2π‘βˆ’π›Ό, and hence 𝛾𝑑(𝐺𝐺)β‰₯2π‘βˆ’π›Ό. Now we set 𝐴 as a 𝑝-set such that |π΄βˆ©π‘‹π‘–|=1, for each 1≀𝑖≀𝑝. Since 𝐴βˆͺ{π‘₯π‘–βˆ£π‘₯π‘–βˆˆπ΄and𝛼+1≀𝑖≀𝑝} is a TDS of 𝐺 of cardinality 2π‘βˆ’π›Ό, we get 𝛾𝑑(𝐺𝐺)=2π‘βˆ’π›Ό.

Corollary 3.3 (see [1]). If 𝑛β‰₯2, then 𝛾𝑑(𝐾𝑛𝐾𝑛)=𝑛.

Proposition 3.4. If 𝐺=𝐾𝑛1,𝑛2,…,𝑛𝑝 is a complete 𝑝-partite graph with 2β‰€π‘˜=𝑛1=β‹―=𝑛𝛼<𝑛𝛼+1≀⋯≀𝑛𝑝, then π›ΎΓ—π‘˜,𝑑𝐺𝐺=𝑝(π‘˜+1)+2π‘˜βˆ’2if𝛼=1,𝑝(π‘˜+1)+𝛼(π‘˜βˆ’1)otherwise.(3.2)

Proof. We discuss 𝛼.
Case 1 (𝛼β‰₯2). It follows by 𝛼β‰₯2 and Lemma 3.1 that, for every π‘˜-tuple total dominating set 𝑆 of 𝐺𝐺, |π‘†βˆ©π‘‹π‘–|β‰₯|π‘†βˆ©π‘‹π‘–|=π‘˜ for 1≀𝑖≀𝛼 and |π‘†βˆ©π‘‹π‘–|β‰₯π‘˜+1 for 𝛼+1≀𝑖≀𝑝. Then π›ΎΓ—π‘˜,𝑑𝐺𝐺β‰₯𝑝(π‘˜+1)+𝛼(π‘˜βˆ’1).(3.3) Now we set ⋃𝐷=(1≀𝑖≀𝛼(𝑋𝑖βˆͺ𝑋𝑖⋃))βˆͺ(𝛼+1≀𝑖≀𝑝𝐷𝑖) such that 𝐷𝑖 is a (π‘˜+1)-subset of 𝑋𝑖, for 𝛼+1≀𝑖≀𝑝. Since 𝐷 is a kTDS of 𝐺𝐺 of cardinality 𝑝(π‘˜+1)+𝛼(π‘˜βˆ’1), we have π›ΎΓ—π‘˜,𝑑(𝐺𝐺)=𝑝(π‘˜+1)+𝛼(π‘˜βˆ’1).Case 2 (𝛼=1). It follows by 𝛼=1 and Lemma 3.1 that, for every kTDS 𝑆 of 𝐺𝐺, 𝑋1βˆͺ𝑋1 is a subset of 𝑆 and also every vertex of 𝑋1βˆͺ𝑋2βˆͺβ‹―βˆͺ𝑋𝑝 is adjacent to at least π‘˜ vertices of π‘†βˆ©(𝑋1βˆͺ𝑋1). Thus either |π‘†βˆ©π‘‹π‘–|=π‘˜+1 for each 2≀𝑖≀𝑝 and βˆ‘2≀𝑖≀𝑝|π‘†βˆ©π‘‹π‘–|β‰₯π‘˜βˆ’1 or ||π‘†βˆ©π‘‹2||||=β‹―=π‘†βˆ©π‘‹π›½||||=π‘˜,π‘†βˆ©π‘‹π›½+1||||=β‹―=π‘†βˆ©π‘‹π‘||=π‘˜+1,(3.4) for some 2≀𝛽≀𝑝. Therefore ||𝑆||β‰₯min{2π‘˜+(π‘˜βˆ’1)+(π‘βˆ’1)(π‘˜+1),2π‘˜+2(π›½βˆ’1)π‘˜+(π‘βˆ’π›½)(π‘˜+1)}=𝑝(π‘˜+1)+2(π‘˜βˆ’1).(3.5) Now we set 𝐷=(𝑋1βˆͺ𝑋1⋃)βˆͺ(2≀𝑖≀𝑝𝐷𝑖)βˆͺ𝐷0 such that 𝐷𝑖 is a (π‘˜+1)-subset of 𝑋𝑖 for 2≀𝑖≀𝑝 and 𝐷0 is a (π‘˜βˆ’1)-subset of 𝑉(𝐺) such that |𝐷0βˆ©π‘‹2|=β‹―=|𝐷0βˆ©π‘‹π‘˜|=1. Since 𝐷 is a kTDS of 𝐺𝐺 of cardinality 𝑝(π‘˜+1)+2π‘˜βˆ’2, we get π›ΎΓ—π‘˜,𝑑(𝐺𝐺)=𝑝(π‘˜+1)+2π‘˜βˆ’2.

Now let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 3β‰€π‘˜+1=𝑛1=β‹―=𝑛𝛼<𝑛𝛼+1≀⋯≀𝑛𝑝, and let 𝑆 be a minimal kTDS of 𝐺𝐺. Then |π‘†βˆ©π‘‹π‘–|β‰₯π‘˜, by Lemma 3.1. We notice that if |π‘†βˆ©π‘‹π‘–|β‰₯π‘˜+2, for some 𝑖, then we may improve 𝑆 and obtain another kTDS 𝑆′ of cardinality |𝑆| such that |π‘†β€²βˆ©π‘‹π‘–|=π‘˜+1 (since every vertex in 𝑋𝑖 (respectively 𝑋𝑖) is adjacent to only one vertex in 𝑋𝑖 (respectively 𝑋𝑖)). Therefore, we may assume that for every minimal kTDS 𝑆 of 𝐺𝐺, we have π‘˜β‰€|π‘†βˆ©π‘‹π‘–|β‰€π‘˜+1.

Now let 𝑆 be a minimal kTDS of 𝐺𝐺, and let 𝐡={𝑖|1≀𝑖≀𝑝,|π‘†βˆ©π‘‹π‘–|=π‘˜} be a set of cardinality 𝛽. We consider the following two cases.

Case 1 (𝛽≠0). In this case, if π‘–βˆˆπ΅, we have |π‘†βˆ©π‘‹π‘–|=|π‘†βˆ©π‘‹π‘–|=π‘˜ such that π‘₯βˆˆπ‘†βˆ©π‘‹π‘– if and only if π‘₯βˆˆπ‘†βˆ©π‘‹π‘–, and |π‘†βˆ©π‘‹π‘–|=π‘˜+1 otherwise. If 𝛽β‰₯2, then ||𝑆||=𝑝(π‘˜+1)+𝛽(π‘˜βˆ’1),(3.6) and if 𝛽=1 and 𝐡={𝑖}, then we have also |π‘†βˆ©(𝑉(𝐺)βˆ’π‘‹π‘–)|=π‘˜. Hence ||𝑆||=𝑝(π‘˜+1)+2π‘˜βˆ’1.(3.7) Comparing (3.6), (3.7) shows that for 𝛽≠0 if 𝑆 is a set of vertices such that π‘†βˆ©π‘‹π‘–={π‘₯π‘–π‘—βˆ£1β‰€π‘—β‰€π‘˜} and π‘†βˆ©π‘‹π‘–={π‘₯π‘–π‘—βˆ£π‘₯π‘–π‘—βˆˆπ‘†βˆ©π‘‹π‘–} for 𝑖=1,2 and |π‘†βˆ©π‘‹π‘–|=π‘˜+1 for 3≀𝑖≀𝑝, then 𝑆 is a minimum kTDS of 𝐺𝐺 and ||𝑆||=𝑝(π‘˜+1)+2π‘˜βˆ’2.(3.8)

Case 2 (𝛽=0). In this case, for each 1≀𝑖≀𝑝 we have |π‘†βˆ©π‘‹π‘–|=π‘˜+1. We continue our discussion in the next subcases.
Subcase 1 (𝛼β‰₯π‘˜+1 or 𝛼=π‘˜β‰€π‘). Then obviously |π‘†βˆ©π‘‰(𝐺)|β‰₯π‘˜. If for 1β‰€π‘–β‰€π‘˜ we consider |π‘†βˆ©π‘‹π‘–|=1, then 𝑆 is a minimum kTDS of 𝐺𝐺 and ||𝑆||=𝑝(π‘˜+1)+π‘˜.(3.9)Subcase 2 (𝛼<π‘˜β‰€π‘). Then obviously |π‘†βˆ©π‘‰(𝐺)|β‰₯π‘˜+1. If we set 𝑆 such that |π‘†βˆ©π‘‹1|=2, and |π‘†βˆ©π‘‹π‘–|=1 when 2β‰€π‘–β‰€π‘˜, then 𝑆 is a minimum kTDS of 𝐺𝐺 and ||𝑆||=𝑝(π‘˜+1)+π‘˜.(3.10)Subcase 3 (𝛼=π‘β‰€π‘˜βˆ’1). Then obviously |π‘†βˆ©π‘‰(𝐺)|β‰₯𝛾×(π‘˜βˆ’1),𝑑(𝐺). If π‘†βˆ©π‘‰(𝐺) is a 𝛾×(π‘˜βˆ’1),𝑑(𝐺)-set, then 𝑆 is a minimum kTDS of 𝐺𝐺, and Proposition B implies ||𝑆||=⎧βŽͺ⎨βŽͺ⎩(𝑝+1)(π‘˜+1)if𝛼=𝑝=π‘˜βˆ’1,𝑝(π‘˜+1)+(π‘˜βˆ’1)π‘ξƒ’π‘βˆ’1if𝛼=𝑝<π‘˜βˆ’1.(3.11)Subcase 4 (𝛼<𝑝<π‘˜). Then obviously |π‘†βˆ©π‘‰(𝐺)|β‰₯π›ΎΓ—π‘˜,𝑑(𝐺). If π‘†βˆ©π‘‰(𝐺) is a π›ΎΓ—π‘˜,𝑑(𝐺)-set, then 𝑆 is a minimum kTDS of 𝐺𝐺, and Proposition B implies ||𝑆||=𝑝(π‘˜+1)+π‘˜π‘ξƒ’.π‘βˆ’1(3.12)

Now let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 4β‰€π‘˜+2≀𝑛1≀𝑛2≀⋯≀𝑛𝑝, and let 𝑆 is a minimal kTDS of 𝐺𝐺. In this case, we may similarly assume that π‘˜β‰€|π‘†βˆ©π‘‹π‘–|β‰€π‘˜+1. Also it can be easily seen that if |π‘†βˆ©π‘‹π‘–|=π‘˜ for some 𝑖, then equality (3.8) holds. Thus let {π‘–βˆ£1≀𝑖≀𝑝,|π‘†βˆ©π‘‹π‘–|=π‘˜}=βˆ…. Then obviously |π‘†βˆ©π‘‰(𝐺)|β‰₯π›ΎΓ—π‘˜,𝑑(𝐺). If we choose a set 𝑆 such that π‘†βˆ©π‘‰(𝐺) is a π›ΎΓ—π‘˜,𝑑(𝐺)-set and |π‘†βˆ©π‘‹π‘–|=π‘˜+1 for 1≀𝑖≀𝑝, then 𝑆 is a minimum kTDS of 𝐺𝐺, and Proposition B implies ||𝑆||=⎧βŽͺ⎨βŽͺ⎩(𝑝+1)(π‘˜+1)if𝑝β‰₯π‘˜+1,(𝑝+1)(π‘˜+1)+1if𝑝=π‘˜,𝑝(π‘˜+1)+π‘˜π‘ξƒ’π‘βˆ’1if𝑝<π‘˜.(3.13) 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 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 3≀𝑛1≀𝑛2≀⋯≀𝑛𝑝. Then 𝛾×2,𝑑(𝐺𝐺)=3𝑝+2.

Proposition 3.6. Let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 4β‰€π‘˜+1=𝑛1=β‹―=𝑛𝛼<𝑛𝛼+1≀⋯≀𝑛𝑝. Then π›ΎΓ—π‘˜,𝑑𝐺𝐺=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘(π‘˜+1)+π‘˜if𝛼=π‘˜β‰€π‘or𝛼β‰₯π‘˜+1(𝑝+1)(π‘˜+1)if𝛼<π‘˜β‰€π‘or𝛼=𝑝=π‘˜βˆ’1,𝑝(π‘˜+1)+min2π‘˜βˆ’2,(π‘˜βˆ’1)π‘π‘βˆ’1ξƒ’ξ‚Όif𝛼=𝑝<π‘˜βˆ’1,𝑝(π‘˜+1)+min2π‘˜βˆ’2,π‘˜π‘π‘βˆ’1ξƒ’ξ‚Όif𝛼<𝑝<π‘˜.(3.14)

Proposition 3.7. Let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 5β‰€π‘˜+2≀𝑛1≀⋯≀𝑛𝑝. Then π›ΎΓ—π‘˜,𝑑𝐺𝐺=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩(𝑝+1)(π‘˜+1)if𝑝β‰₯π‘˜+1(𝑝+1)(π‘˜+1)+1if𝑝=π‘˜β‰₯4,16if𝑝𝑝=π‘˜=3,(π‘˜+1)+min2π‘˜βˆ’2,π‘˜π‘π‘βˆ’1ξƒ’ξ‚Όif𝑝<π‘˜.(3.15)

We now determine the π‘˜-tuple total domination number of the complementary prism 𝐢𝑛𝐢𝑛, where 1β‰€π‘˜β‰€3=𝛿(𝐢𝑛𝐢𝑛). Here we assume that 𝑉(𝐢𝑛𝐢𝑛)=𝑉(𝐢𝑛)βˆͺ𝑉(𝐢𝑛), 𝑉(𝐢𝑛)={π‘–βˆ£1≀𝑖≀𝑛}, and 𝐸(𝐢𝑛)={(𝑖,𝑖+1)∣1≀𝑖≀𝑛}. Proposition D implies that 𝛾×3,𝑑(𝐢𝑛𝐢𝑛)=2𝑛. In many references, for example, in [1], it can be seen that, for 𝑛β‰₯3,𝛾𝑑𝐢𝑛=⎧βŽͺ⎨βŽͺ⎩2ξ‚œπ‘›4if2ξ‚œπ‘›π‘›β‰’1(mod4),4ξ‚βˆ’1if𝑛≑1(mod4),(3.16) and trivially we can prove𝛾𝑑𝐢𝑛=⎧βŽͺ⎨βŽͺ⎩4if3𝑛=4,if2𝑛=5,if𝑛β‰₯6.(3.17) Hence Theorem 2.1 implies that𝛾𝑑𝐢𝑛+2≀𝛾×2,𝑑𝐢𝑛𝐢𝑛≀𝑛+2,(3.18) where 𝑛β‰₯6, and also Theorem 2.2 implies that𝑛≀𝛾×2,𝑑𝐢𝑛𝐢𝑛≀𝑛+𝛾×2,𝑑𝐢𝑛,(3.19) where 𝑛β‰₯5. In chain (3.19) we need to calculate 𝛾×2,𝑑(𝐢𝑛), which is done by the next proposition.

Proposition 3.8. If C𝑛 is a cycle of order nβ‰₯5, then 𝛾×2,𝑑𝐢𝑛=⎧βŽͺ⎨βŽͺ⎩5if4𝑛=5,if36≀𝑛≀8,if𝑛β‰₯9.(3.20)

Proof. Proposition C implies that 𝛾×2,𝑑(𝐢𝑛)β‰₯3. If 𝑛β‰₯9, then, for each 1≀𝑖≀𝑛, the set {𝑖,𝑖+3,𝑖+6} is a DTDS of 𝐢𝑛 and so 𝛾×2,𝑑(𝐢𝑛)=3. If 6≀𝑛≀8, then it can be easily verified that 𝛾×2,𝑑(𝐢𝑛)β‰₯4. Now since {1,3,4,6} and {1,2,4,6} are double total dominating sets of 𝐢𝑛, where 𝑛=6 and 𝑛=7,8, respectively, we get 𝛾×2,𝑑(𝐢𝑛)=4. Finally if 𝑛=5, then 𝐢5 is 2-regular and Proposition D implies 𝛾×2,𝑑(𝐢5)=5.

Proposition 3.9. If 𝑛β‰₯5, then 𝛾×2,𝑑(𝐢𝑛𝐢𝑛)=𝑛+2.

Proof. Let 𝑛β‰₯5. equalities (3.18), (3.19) and Propositions C and 3.8 imply ξ‚†ξ‚œmax𝑛,4π‘›π‘›βˆ’2≀𝛾×2,𝑑𝐢𝑛𝐢𝑛≀𝑛+2.(3.21) If 𝑛=5, then max{𝑛,⌈4𝑛/(π‘›βˆ’2)βŒ‰}=⌈4𝑛/(π‘›βˆ’2)βŒ‰=7=𝑛+2, and so 𝛾×2,𝑑(𝐢𝑛𝐢𝑛)=𝑛+2. Thus we assume 𝑛β‰₯6. Then max{𝑛,⌈4𝑛/(π‘›βˆ’2)βŒ‰}=𝑛 and hence 𝑛≀𝛾×2,𝑑(𝐢𝑛𝐢𝑛)≀𝑛+2. Now let 𝑆 be a 𝛾×2,𝑑(𝐢𝑛𝐢𝑛)-set. If 𝑉(𝐢𝑛)βŠ†π‘†, then 𝑆=𝑉(𝐢𝑛)βˆͺ{π‘₯,𝑦}, for some two adjacent vertices π‘₯,π‘¦βˆˆπ‘‰(𝐢𝑛), and so 𝛾×2,𝑑(𝐢𝑛𝐢𝑛)=𝑛+2. Thus we assume 𝑉(𝐢𝑛)ΜΈβŠ†π‘†. Without loss of generality, let 3βˆ‰π‘†. Since |π‘†βˆ©{2,4,3}|β‰₯2, we continue our proof in the following two cases.Case 1 ({2,4}βŠ†π‘†). Then 1,5,2,4βˆˆπ‘†. We note that, for every 5β‰€π‘–β‰€π‘›βˆ’1, π‘†βˆ©{𝑖,𝑖+1}β‰ βˆ…. This implies |𝑆|β‰₯(π‘›βˆ’1βˆ’4)+6=𝑛+1, and since 3 must be dominated by π‘†βˆ©π‘‰(𝐢𝑛), we have |π‘†βˆ©π‘‰(𝐢𝑛)|β‰₯4. If π‘›βˆ‰π‘†, then 1βˆˆπ‘† and so |𝑆|β‰₯𝑛+1+|{1}|=𝑛+2. Let π‘›βˆˆπ‘†. If π‘›βˆˆπ‘†, again |𝑆|β‰₯𝑛+1+|{𝑛}|=𝑛+2. But π‘›βˆ‰π‘† implies π‘›βˆ’1βˆˆπ‘†. Let ={π‘–βˆˆπ‘†βˆ£5β‰€π‘–β‰€π‘›βˆ’1andπ‘–βˆˆπ‘†}. The condition |π‘†βˆ©π‘‰(𝐢𝑛)|β‰₯4 implies ||β‰₯2. Therefore for at least one vertex 5≀π‘₯β‰€π‘›βˆ’1, {π‘₯,π‘₯+1}βŠ†π‘† and hence |𝑆|β‰₯𝑛+|{π‘₯,π‘₯+1}|=𝑛+2.Case 2 ({4,3}βŠ†π‘† (similarly {2,3}βŠ†π‘†)). Case 1 implies 2βˆ‰π‘†. Then 1,2,4,1,4,5βˆˆπ‘†. Again we see that, for every 5β‰€π‘–β‰€π‘›βˆ’2, π‘†βˆ©{𝑖,𝑖+1}β‰ βˆ… and so |𝑆|β‰₯(π‘›βˆ’2βˆ’4)+8=𝑛+2.Therefore, in the previous all cases, we proved that 𝛾×2,𝑑(𝐢𝑛𝐢𝑛)β‰₯𝑛+2 and chain (3.21) implies 𝛾×2,𝑑(𝐢𝑛𝐢𝑛)=𝑛+2.

Corollary 3.10. If 𝑛β‰₯5, then 𝛾×2,𝑑𝐢𝑛𝐢𝑛=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π›ΎΓ—2,𝑑𝐢𝑛+𝛾×2,π‘‘ξ‚€πΆπ‘›ξ‚βˆ’1if𝛾𝑛β‰₯9,Γ—2,𝑑𝐢𝑛+𝛾×2,π‘‘ξ‚€πΆπ‘›ξ‚βˆ’2if𝛾6≀𝑛≀8,Γ—2,𝑑𝐢𝑛+𝛾×2,π‘‘ξ‚€πΆπ‘›ξ‚βˆ’3if𝑛=5.(3.22)

Now we determine the exact amount of 𝛾𝑑(𝐢𝑛𝐢𝑛) for 𝑛β‰₯3. Obviously 𝛾𝑑(𝐢3𝐢3)=|𝑉(𝐢3)|=3. In the next proposition we calculate it when 𝑛β‰₯4.

Proposition 3.11. Let 𝑛β‰₯4. Then 𝛾𝑑𝐢𝑛𝐢𝑛=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩2ξ‚œπ‘›4+2if2ξ‚œπ‘›π‘›β‰‘0(mod4),4+1if2ξ‚œπ‘›π‘›β‰‘3(mod4),4otherwise.(3.23)

Proof. Theorem 2.2 with equalities (3.16) and (3.17) implies 4≀𝛾𝑑𝐢4𝐢4≀6,4≀𝛾𝑑𝐢5𝐢5≀8,(3.24) and if 𝑛β‰₯6 and 𝑛≒1(mod4), then 2ξ‚œπ‘›4ξ‚β‰€π›Ύπ‘‘ξ‚€πΆπ‘›πΆπ‘›ξ‚ξ‚œπ‘›β‰€24+2,(3.25) and if 𝑛β‰₯6 and 𝑛≑1(mod4), then 2ξ‚œπ‘›4ξ‚βˆ’1β‰€π›Ύπ‘‘ξ‚€πΆπ‘›πΆπ‘›ξ‚ξ‚œπ‘›β‰€24+1.(3.26) If 𝑛=4 and 𝑛=5, then the sets {1,2,1,2} and {1,1,4,4} are total dominating sets of 𝐢𝑛𝐢𝑛, respectively. Hence chain (3.24) implies 𝛾𝑑(𝐢𝑛𝐢𝑛)=4 for 𝑛=4,5. Now we assume 𝑛β‰₯6. For 𝑛≑2(mod4), since the sets {1,1,4,4} and {1,1,4,4}βˆͺ{7+4𝑖,8+4π‘–βˆ£0β‰€π‘–β‰€βŒˆπ‘›/4βŒ‰βˆ’3} are two total dominating sets of 𝐢𝑛𝐢𝑛 of cardinality 2βŒˆπ‘›/4βŒ‰, where 𝑛=6 and 𝑛>6, respectively, we have 𝛾𝑑(𝐢𝑛𝐢𝑛)=2βŒˆπ‘›/4βŒ‰, by chain (3.25). Now let 𝑛≒2(mod4). We assume that 𝑆 is a TDS of 𝐢𝑛𝐢𝑛. Obviously π‘†βˆ©π‘‰(𝐢𝑛)β‰ βˆ…. If |π‘†βˆ©π‘‰(𝐢𝑛)|=1 and π‘†βˆ©π‘‰(𝐢𝑛)={1}, then 1,2,π‘›βˆˆπ‘†, and hence |π‘†βˆ©π‘‹|β‰₯2⌈|𝑋|/4βŒ‰=2⌈(π‘›βˆ’5)/4βŒ‰, where 𝑋=𝑉(𝐢𝑛)βˆ’{1,2,3,π‘›βˆ’1,𝑛}. This implies ||𝑆||=||||⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩2ξ‚œπ‘›π‘†βˆ©π‘‹+4β‰₯4+2if2ξ‚œπ‘›π‘›β‰‘0(mod4),4+1if2ξ‚œπ‘›π‘›β‰‘3(mod4),4if𝑛≑1(mod4).(3.27) Now let |π‘†βˆ©π‘‰(𝐢𝑛)|=𝛼β‰₯2. If 𝑛≑0,1(mod4), then ||ξ€·πΆπ‘†βˆ©π‘‰π‘›ξ€Έ||β‰₯⎧βŽͺ⎨βŽͺ⎩2ξ‚žπ‘›βˆ’π›Ό4ξ‚Ÿif2ξ‚žπ‘›β‰‘π›Ό(mod4),π‘›βˆ’π›Ό4ξ‚Ÿ+1otherwise,(3.28) and if 𝑛≑3(mod4), then ||ξ€·πΆπ‘†βˆ©π‘‰π‘›ξ€Έ||β‰₯⎧βŽͺ⎨βŽͺ⎩2ξ‚œπ‘›βˆ’π›Ό4ξ‚βˆ’1if2ξ‚œπ‘›β‰‘π›Ό+1(mod4),π‘›βˆ’π›Ό4otherwise.(3.29) It can be calculated that ||𝑆||=||ξ€·πΆπ‘†βˆ©π‘‰π‘›ξ€Έ||⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩2ξ‚œπ‘›+𝛼β‰₯4+2if2ξ‚œπ‘›π‘›β‰‘0(mod4),4+1if2ξ‚œπ‘›π‘›β‰‘3(mod4),4if𝑛≑1(mod4).(3.30) Then by chains (3.25) and (3.26) we have π›Ύπ‘‘ξ‚€πΆπ‘›πΆπ‘›ξ‚ξ‚œπ‘›=24+2if𝑛≑0(mod4),(3.31)2ξ‚œπ‘›4ξ‚β‰€π›Ύπ‘‘ξ‚€πΆπ‘›πΆπ‘›ξ‚ξ‚œπ‘›β‰€24+1if𝑛≑1(mod4),(3.32)2ξ‚œπ‘›4+1β‰€π›Ύπ‘‘ξ‚€πΆπ‘›πΆπ‘›ξ‚ξ‚œπ‘›β‰€24+2if𝑛≑3(mod4).(3.33) If 𝑛≑1(mod4), then the sets {1,1,4,4,7,7} and {1,1,4,4,7,7}βˆͺ{10+4𝑖,11+4π‘–βˆ£0β‰€π‘–β‰€βŒˆπ‘›/4βŒ‰βˆ’4} are total dominating sets of 𝐢𝑛𝐢𝑛 of cardinality 2βŒˆπ‘›/4βŒ‰ when 𝑛=9 and 𝑛>9, respectively. Hence 𝛾𝑑(𝐢𝑛𝐢𝑛)=2βŒˆπ‘›/4βŒ‰, by chain (3.32). If also 𝑛≑3(mod4), the sets {1,1,4,4,π‘›βˆ’1} and {1,1,4,4,π‘›βˆ’1}βˆͺ{7+4𝑖,8+4π‘–βˆ£0β‰€π‘–β‰€βŒˆπ‘›/4βŒ‰βˆ’3} are total dominating sets of 𝐢𝑛𝐢𝑛 of cardinality 2βŒˆπ‘›/4βŒ‰+1 when 𝑛=7 and 𝑛>7, respectively. Hence 𝛾𝑑(𝐢𝑛𝐢𝑛)=2βŒˆπ‘›/4βŒ‰+1, by chain (3.33).

Finally we determine the π‘˜-tuple total domination number of the complementary prism 𝑃𝑛𝑃𝑛, where 1β‰€π‘˜<2=𝛿(𝑃𝑛𝑃𝑛). We recall that 𝑉(𝑃𝑛𝑃𝑛)=𝑉(𝑃𝑛)βˆͺ𝑉(𝑃𝑛), 𝑉(𝑃𝑛)={π‘–βˆ£1≀𝑖≀𝑛}, and 𝐸(𝑃𝑛)={π‘–π‘—βˆ£1β‰€π‘–β‰€π‘›βˆ’1,𝑗=𝑖+1}. In many references, for example, in [1], it can be seen that, for 𝑛β‰₯2,𝛾𝑑𝑃𝑛=⎧βŽͺ⎨βŽͺ⎩2ξ‚œπ‘›4if2ξ‚œπ‘›π‘›β‰’1(mod4),4ξ‚βˆ’1if𝑛≑1(mod4),(3.34) and trivially 𝛾𝑑(𝑃𝑛)=|{1,𝑛}|=2, where 𝑛β‰₯4. Therefore, by Theorems 2.1 and 2.2, for 𝑛β‰₯4, we have the following chain:𝛾𝑑𝑃𝑛≀𝛾𝑑𝑃𝑛𝑃𝑛≀𝛾𝑑𝑃𝑛+2≀𝛾×2,𝑑𝑃𝑛𝑃𝑛≀𝑛+2.(3.35) It can be easily proved that 𝛾𝑑(𝑃𝑛𝑃𝑛)=𝑛, where 𝑛=2,3. Next proposition calculates 𝛾𝑑(𝑃𝑛𝑃𝑛) when 𝑛β‰₯4.

Proposition 3.12. Let 𝑛β‰₯4. Then 𝛾𝑑𝑃𝑛𝑃𝑛=⎧βŽͺ⎨βŽͺ⎩2ξ‚œπ‘›βˆ’24+1if2ξ‚œπ‘›β‰‘3(mod4),π‘›βˆ’24+2otherwise.(3.36)

Proof. Let 𝐷 be a 𝛾𝑑-set of the induced path 𝑃𝑛[𝑉(𝑃𝑛)βˆ’{1,𝑛}] of 𝑃𝑛. Since 𝐷βˆͺ{1,𝑛} is a TDS of 𝑃𝑛𝑃𝑛, we have 𝛾𝑑𝑃𝑛𝑃𝑛≀|||𝐷βˆͺ1,𝑛|||=⎧βŽͺ⎨βŽͺ⎩2ξ‚œπ‘›βˆ’24+1if2ξ‚œπ‘›β‰‘3(mod4),π‘›βˆ’24+2otherwise.(3.37) Let 𝑛≑2(mod4). Then chains (3.34), (3.35), (3.37) imply 𝛾𝑑(𝑃𝑛𝑃𝑛)=2⌈(π‘›βˆ’2)/4βŒ‰+2. Since 2βŒˆπ‘›/4βŒ‰=2⌈(π‘›βˆ’2)/4βŒ‰+2. Now let 𝑛≒2(mod4), and let 𝑆 be a TDS of 𝑃𝑛𝑃𝑛. Obviously π‘†βˆ©π‘‰(𝑃𝑛)β‰ βˆ…. In all cases, (i) |π‘†βˆ©π‘‰(𝑃𝑛)|=1 and π‘†βˆ©{1,𝑛}β‰ βˆ…, (ii) |π‘†βˆ©π‘‰(𝑃𝑛)|=1 and π‘†βˆ©{1,𝑛}=βˆ…, and (iii) |π‘†βˆ©π‘‰(𝑃𝑛)|β‰₯2, then similar to the proof of Proposition 3.11, it can be verified that ||𝑆||β‰₯⎧βŽͺ⎨βŽͺ⎩2ξ‚œπ‘›βˆ’24+1if2ξ‚œπ‘›β‰‘3(mod4),π‘›βˆ’24+2otherwise.(3.38) 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 𝐺∈{𝑃𝑛,𝐢𝑛} with order 𝑛β‰₯5, then 𝛾𝑑𝐺𝐺=⎧βŽͺ⎨βŽͺβŽ©π›Ύπ‘‘(𝐺)if𝛾𝑛≑2(mod4),𝑑(𝐺)+2if𝛾𝑛≑0(mod4),𝑑(𝐺)+1otherwise.(3.39)

4. Problems

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

Proposition 4.1. (i) Let 𝐺 be a cycle or a path of order 𝑛β‰₯4. Then max{𝛾𝑑(𝐺),𝛾𝑑(𝐺)}=𝛾𝑑(𝐺𝐺) if and only if 𝑛≑2(mod4).
(ii) Let 𝐺 be a cycle of order 𝑛β‰₯5 or a path of order 𝑛β‰₯4. Then 𝛾𝑑(𝐺𝐺)=𝛾𝑑(𝐺)+𝛾𝑑(𝐺) if and only if 𝑛≑0(mod4).
(iii) Let 𝐢𝑛 be a cycle of order 𝑛β‰₯5. Then 𝛾maxΓ—2,𝑑𝐢𝑛,𝛾×2,𝑑𝐢𝑛<𝛾×2,𝑑𝐢𝑛𝐢𝑛<𝛾×2,𝑑𝐢𝑛+𝛾×2,𝑑𝐢𝑛.(4.1)
(iv) Let 𝐢𝑛 be a cycle of order 𝑛β‰₯5. Then 𝛾𝑑𝐢𝑛+𝛾𝑑𝐢𝑛<𝛾×2,𝑑𝐢𝑛𝐢𝑛𝛾=𝑛+min𝑑𝐢𝑛,𝛾𝑑𝐢𝑛.(4.2)

Therefore it is natural that we state the following problem.

Problem 1. Characterize graphs 𝐺 with(1)π›ΎΓ—π‘˜,𝑑(𝐺𝐺)=π›ΎΓ—π‘˜,𝑑(𝐺)+π›ΎΓ—π‘˜,𝑑(𝐺), (2)π›ΎΓ—π‘˜,𝑑(𝐺)=max{π›ΎΓ—π‘˜,𝑑(𝐺),π›ΎΓ—π‘˜,𝑑(𝐺)}.

References

  1. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs, vol. 208 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1998.
  2. T. W. Haynes, M. A. Henning, and L. C. van der Merwe, β€œDomination and total domination in complementary prisms,” Journal of Combinatorial Optimization, vol. 18, no. 1, pp. 23–37, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. T. W. Haynes, M. A. Henning, P. J. Slater, and L. C. van der Merwe, β€œThe complementary product of two graphs,” Bulletin of the Institute of Combinatorics and its Applications, vol. 51, pp. 21–30, 2007. View at Zentralblatt MATH
  4. M. A. Henning and A. P. Kazemi, β€œk-tuple total domination in graphs,” Discrete Applied Mathematics, vol. 158, no. 9, pp. 1006–1011, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. M. A. Henning and A. P. Kazemi, β€œk-tuple total domination in cross products of graphs,” Journal of Combinatorial Optimization. View at Publisher Β· View at Google Scholar
  6. A. Khodkar, D. A. Mojdeh, and A. P. Kazemi, β€œDomination in Harary graphs,” Bulletin of the Institute of Combinatorics and its Applications, vol. 49, pp. 61–78, 2007. View at Zentralblatt MATH