Isolate Domination of Some Special Graphs

JOURNAL OF HIGHER EDUCATION RESEARCH DISCIPLINES ISSN: (ONLINE) 2546-0579 Tangub City, Misamis Occidental, Philippines (PRINT) 2546-0560 December 2017

ISOLATE DOMINATION OF SOME SPECIAL GRAPHS

Shaleema A. Arriola [email protected] Basilan State College

ABSTRACT

A dominating set of is called an isolate dominating set of if the subgraph induced by has an isolated vertex. In this paper, we investigate the isolate domination푆 퐺 of some special graphs such as cocktail퐺 party graph, crown〈푆〉 graph, gear푆 graph, and jump graph. Characterizations and exact values of isolate domination number were also derived for these graphs.

Keywords: domination, isolate domination, cocktail party graph, crown graph, gear graph, jump graph

1.0 Introduction

Throughout this paper, we only consider a simple graph (a graph without loop or multiple edges). For a graph , we denote ( ) and ( ) as vertex set and edge set of , respectively. Two vertices and are said to be neighbor or adjacent (denoted by ) if ( 퐺). The set of푉 all퐺 neighbors퐸 퐺 of is denoted by ( ) and it is퐺 called the open neighborhood 푢of , that푣 is, ( ) = { ( ): ( )}. The closed neighborhood푢푣 푢푣 ∈ 퐸 of퐺 , denoted by [ ], is given푢 by [ ] = 푁(푢) { }. An induced subgraph of (simply,푢 subgraph푁 푢of ) is푥 ∈a 푉graph퐺 푥푢with∈ vertices퐸 퐺 less than or equal to ( ) such푢 that adjacencies푁 and푢 non-adjacencies푁 of푢 any two푁 푢 vertices∪ 푢 in are preserved. A subgraph퐺 of induced by 퐺 is denoted by . The complement of a graph 푉, 퐺denoted by , is a graph of the same vertices as such that any two퐺 vertices are adjacent if and only퐺 if they are not푆 adjacent in . Any〈푆〉 undefined terms not specified퐺 in this paper, may퐺̅ see Harary (1969). 퐺 퐺 We say that a subset ( ) is a dominating set of if for all vertex , there exists a vertex ( )\ such that ( ) . The minimum cardinality of the dominating set푆 ⊆ of푉 퐺 is called a domination number퐺 of , denoted 푥by∈ 푆( ) Haynes et al (1998). If푣 a∈ subgraph푉 퐺 푆 induced푢푣 by∈ a퐸 do퐺minating set of has an isolated vertex, then we say퐺 that is an isolate dominating set 퐺of . The minimum훾 퐺 cardinality of an isolate dominating〈푆〉 set is called isolate domination푆 퐺 number, denoted by ( ). The concept of푆 isolate domination was introduced퐺 by Hamid & Balamurugam (2013) and further studied by Arriola (2015). 훾0 퐺 Let us take some examples to illustrate the concept of domination and isolate domination. Consider the graph as shown in Figure 1. Sets = { } and = { , } are both dominating sets, and also both an isolate dominating sets. 퐺 푆1 푢1 푆2 푢1 푢2 23

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Since the cardinality of is less than the cardinality of , it follows that ( ) = 1. 1 2 0 푆 푆 훾 퐺 A complete graph is a graph on which every pair of vertices is connected by an edge. The complete graph with vertices is denoted by . Figure 2 shows a complete graph with 5 vertices. A dominating set or isolate dominating set of a certain graph need not be unique. For푛 instance, the set of any퐾푛 single vertex of a complete graph constitute an isolate dominating set.

Figure 2: Complete Graph with 5 vertices Figure 1 Simple Graph

2.0 Main Results

In this section, we discuss some special graphs and illustrate these concepts through examples. Characterizations of the isolate domination and some exact values of its corresponding isolate domination number are also discussed in detailed.

2.1 Cocktail Party Graph

Biggs (1993) introduced the concept of cocktail party graph.

Definition 2.1.1 Let and be two sets of vertices, respectively, such that every vertex in has1 a vertex2 pair in . If such that each vertex in is connected1 푉 by푉 an edge to every2 vertex푛1 in 2 except the paired ones, then 푛 we1 say that푉 such graph is a cocktail푉 party〈 푉graph〉 ≅,〈 denoted푉 〉2≅ 퐾 by . 〈푉 〉 〈푉 〉 Consider the graph shown in Figure 3. This graph illustrates�퐿�푛� a cocktail party graph since both subgraphs { , , , } and { , , , } are complete, and corresponding paired vertices are not connected by an edge. The following sets are 〈 푢1 푢2 푢3 푢4 〉 〈 푣1 푣2 푣3 푣4 〉

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the isolate dominating sets: = { , } , = { , } , = { , } , and = { , }. 1 1 1 2 2 2 3 3 3 푆 푢 푣 푆 푢 푣 푆 푢 푣 푆4 푢4 푣4

Figure 3: Cocktail Party Graph

Theorem 2.1.2 Any isolate dominating set of a cocktail party graph of order 2 has size two. In particular, ( ) = 2. 푛 퐿� 푛 0 푛 Proof: Suppose is훾 an�퐿 isolate�� dominating set of such that | | 2. Suppose further that | | = 1 and let = { }. All vertices are dominated by 푛 except its vertex pair. Thus,푆 is not a dominating set of퐿� �� , a contradiction.푆 ≠ Also, suppose | | 푆3 . If 푆 (similarly,푢 if ), then has no푢 푛 isolated vertex, a contradiction.푆 If 1 and 퐿�2�� , then has no isolated vertex, a푆 contradiction.≥ 푆 ⊆ Thus,푉 1( ) = 2. 푆 ⊆ 푉 2 푆 푆 ∩ 푉 ≠ ∅ 푆 ∩ 푉 ≠ ∅ 푆 0 푛 2.2 Crown Graph 훾 �퐿�� ∎

The crown graph is similar to the cocktail party graph but differs only on its subgraphs. Both subgraphs and of a crown graph are complement of complete graphs. This crown graph1 was2 first studied by Brouwer et. al. (1989) and showed that crown graph is distance푉 —푉transitive and isomorphic to the complement of the rook graph.

Definition 2.2.1 Let and be two sets of vertices, respectively, such that every vertex in has a vertex1 pair2 in . If such that each vertex in is connected1 by푉 an edge푉 to every2 vertex1 푛 in 2 except the paired ones, then we1 say ��푛� that푉 such graph is a crown 푉graph,〈 denoted푉 〉 ≅ 〈 푉by〉2 ≅. 퐾 〈푉 〉 〈푉 〉0 푆푛

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Figure 4 is an example of a crown graph. The corresponding isolate domination number is also 2 and the isolate dominating set for this graph is characterize in the next theorem. The following remark is consequent to the given definition of a crown graph.

Figure 4: Crown Graph

Remark 2.2.2 The isolate dominating set of any crown graph has at least two elements.

Theorem 2.2.3 Let be a crown graph. Then a set ( ) is an isolate dominating set of if and only0 if any of the following conditions hold:0 푛 푛 0 푆 i. { }, where 푆 ⊆ 푉and푆 its paired vertex in is in 푛 푆 such that1 = whenever2 ; 1 ii. 푆 ⊆ {푉 }∪ 푥 , where 푥1 ∈ 푉 and its paired vertex in 푉 is in 푆 such that 2 푆= 푉 whenever1 푥 ∉ 푆. 2 푆 ⊆ 푦 ∪ 푉 2푦 ∈ 푉 푉 푆 Proof: Suppose is an isolate푆 dominating푉 set푦 of∉ 푆 . Let and be the vertex partition of as specified in Definition 2.2.1. If0 | | = 21, the proof2 is 푛 straightforward. So, we0 may푆 assume that | | 3 and consider푆 the following푉 푉 cases: 푛 푆 푆 Case 1: = (similarly, = 푆) ≥ Then, 1 . If , the2n there exists at least one element not in . This implies푆 ∩ 푉that this∅ 2 particular 푆element2∩ 푉 is∅ not dominated by any element of since is an empty푆 graph.⊆ 푉 Hence,푆 ≠ 푉= . 푆 2 2 푆 푉Case 2: and 푆 푉 If | 1 | 2 and | 2 | 2, then is a connected graph which is a contradiction푆 ∩ 푉 to≠ 1our∅ assumption푆 ∩ 푉 ≠ that2∅ is an isolate dominating set. Thus, either | | =푆1∩ or푉 | ≥ | = 1푆. ∩This푉 ≥follows that〈푆〉 { }, where or { 1} . 2 푆 1 2 푆 ∩ 푉 2 푆 ∩ 푉 푆 ⊆ 푉 ∪ 푥 푥 ∈ 푉 푆 ⊆ 푦 ∪ 푉

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For the converse, assume that the given conditions hold. If = , then clearly is an isolate dominating set. Also, if = , then is an 1isolate dominating set. Suppose { }, where and2 a paired vertex푆 푉 of is 푆in . Then vertex dominate1 any vertices푆 2in 푉 while 푆’ dominate any1 vertices in and is an푆 isolated⊆ 푉 ∪ vertex푥 in 푥. Similarly,∈ 푉 1 for { } 푥, ′where∈ 푉 푥 and푆 a2 paired vertex 푥 of is in . 푉 푥 2 1 푉 푥′ 2 〈푆〉 푆 ⊆ 푦 ∪ 푉 푦Corollary∈ 푉 2.2.4 푦′ ∈ 푉 푦 푆 ∎ If is a crown graph of order 2 , then ( ) = 2. 0 0 푛 0 푛 Proof푆 : Follows from Theorem 2.2.3.푛 훾 푆

2.3 Gear Graph ∎

Kirlangic (2009) derived some results about the rupture degree of gear graphs and further considered using the neighbor rupture degree by Bacak-Turan & Demirtekin (2017).

Definition 2.3.1 A gear graph, denoted by is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph 푛 (Brandstadt et. al, 1999). 퐺 푊푛 For a gear graph , we denote as the central vertex, = ( ) be the set of all neighbors of 푛 , and be the푐 set of all vertices not푤 adjacent퐺푛 to푐 . As an example, the퐺 first graph shown푥 in Figure 4 is an example푉 푁 of 푥wheel graph while the second graph푥푐 is an example푉��푤� of gear graph. 푥푐

Figure 5: Wheel Graph and Gear Graph

Theorem 2.3.2

Let be a gear graph of order 2 + 1 with 2. Then the set ( ) is an isolate dominating set of if and only if either of the following holds: 푛 푛 퐺 i. { } such푛 that 푛 ≥ ( ) 푆and⊆ 푉there퐺 푛 exists 퐺 with ( ) = ; or 푆 ⊆ 푆1 ∪ 푆2 ∪ 푥푐 �푉�푤� ⊆ 푁퐺푛 푆1 ∪ 푆2 푢 ∈ 푆2 푁퐺푛 푢 ∩ 푆1 ∅

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ii. = such that , ( ) ( ) = ( ) , and either there exists with ( ) = or 1 2 1 퐺푛 1 퐺푛 2 푛 푆there푆 exists∪ 푆 with 푆 ≠ ∅ 푁 (푆) ∪ 푁 = 푆, 푉 퐺 1 퐺푛 2 . 푢 ∈ 푆 푁 푢 ∩ 푆 ∅ where and 2 퐺푛 1 푣 ∈ 푆 푁 푢 ∩ 푆 ∅ 1 푤 2 푤 푆Proof:⊆ 푉 Suppose푆 ⊆ �푉 �is� an isolate dominating set of . Let be the central vertex of . Consider the following cases: 푛 푐 푆 퐺 푥 푛 Case 1: 퐺 Since is a dominating set, every element of has neighbor in or it is 푐 in . Take푥 ∈ 푆be the set of some elements of that are in and be the set of 푤 some elements푆 of that are neighbors of that are푉� in . This implies푆 that 2 푤 1 푆( ) 푆 contain all elements of , that 푉is,� ( 푆) 푆. Since is an 푤 2 isolate dominating set,푉 there exists at least one푆 vertex in 푆 = such 퐺푛 1 2 �푤 �푤 퐺푛 1 2 푁that all푆 its∪ neighbor푆 are not in . Thus, 푉if , then푉 ⊆ 푁( )푆 ∪ 푆= . 푆 푤 2 푉� ∩ 푆 푆 2 퐺푛 1 Case 2: 푆 푢 ∈ 푆 푁 푢 ∩ 푆 ∅ Since is a dominating set, then at least one element of must be in . 푐 Take 푥=∉ 푆 and then . Moreover, since is a dominating set, 푤 ( ) = ( 푆 ). Take = \ . Then ( ) = ( ) (푉 ). Also, since푆 1 푤 1 is an푆 isolate푉 ∩ dominating푆 set,푆 ≠there∅ exists an element 푆in , say , such that 퐺푛 푛 2 1 푛 퐺푛 1 퐺푛 2 푁 (푆) 푉=퐺 . If 푆 , then푆 푆 ( ) 푉 퐺 = .푁 If 푆 ∪,푁 then푆 ( ) = 푆. 푆 푥 퐺푛 1 퐺푛 2 2 퐺푛 1 푁 푥 ∩ 푆 ∅ 푥 ∈ 푆 푁 푥 ∩ 푆 ∅ 푥 ∈ 푆 푁 푥 ∩ 푆 ∅ The proof of the converse is straightforward.

∎ Corollary 2.3.3 If is a gear graph of order 2 + 1, then ( ) = . 2푛 푛 0 푛 Proof:퐺 Since the gear graph without푛 the central훾 퐺 vertex� 3 �is isomorphic to a cycle. We can choose first the vertex adjacent to the central vertex and then add every other three vertices to the set. Using Theorem 2.3.2, this set constitute an isolate dominating set for the gear graph. Formal argument is straightforward.

2.4 Jump Graph ∎

Given any graph, the line graph is simply changing its edges and vertices into vertices and edges, respectively, such that adjacency is preserved. The jump graph may directly derived from the given graph or through the complement of its corresponding line graph. Maralabhavi et al (2013) particularly studied the graph theoretic properties of the domination number of jump graph.

Definition 2.4.1

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The jump graph ( ) of a graph is the graph define on ( ) and in which two vertices are adjacent if and only if they are not adjacent in . 퐽 퐺 퐺 퐸 퐺 For example, given a graph (see the first graph) in Figure퐺 5. The corresponding jump graph of is shown in second graph of Figure 5. Observe that all edges of become vertices of ( )퐺 and its two vertices in ( ) are adjacent if they are not edge adjacent in 퐺. Jump graph is isomorphic to the complement of a line graph. 퐺 퐽 퐺 퐽 퐺 퐺

Figure 6: Graph G and its jump graph J(G)

Theorem 2.4.2 Let be any graph of size 1. Then ( ( )) = 1 if and only if contains a component . 0 퐺 푚 ≥ 훾 퐽 퐺 퐺 Proof: Suppos푃 2 ( ( )) = 1 and let = { } be the isolate dominating set of ( ). This implies that a vertex dominates all other vertices in ( ). Since 0 corresponds to an edge훾 in 퐽 퐺and has no adjacent푆′ edges,푒 is an isolated edge in . Thus,퐽 퐺 contains a component . The푒 converse is obvious. 퐽 퐺 푒 퐺 푒 퐺 2 Theorem퐺 2.4.3 푃 ∎ Let be a graph of size 3. Then ( ( )) = 2 if and only if there exists ( ) such that ( ) = 2. 0 퐺 푚 ≥ 훾 퐽 퐺 퐺 푢 Proof:∈ 푉 퐺 Suppose 푑푒푔( ( )푢) = 2 and let = { , } . Then and are adjacent edges such that either or is not adjacent′ to any edges in . This 0 1 2 1 2 implies that there is no 훾other퐽 퐺 edge adjacent to푆 both 푒 푒and . Thus,푒 if 푒is the 1 2 common vertex connecting 푒 and 푒 , then ( ) = 2 . The converse퐺 is 1 2 straightforward. 푒 푒 푢 푒1 푒2 푑푒푔퐺 푢 Corollary 2.4.4 ∎ Let { , , } for 3. Then ( ( )) = 2.

Proof:퐺 ∈ Follows푃푛 퐶푛 from퐹푛 Theorem푛 ≥ 2.4.3. 훾0 퐽 퐺

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Theorem 2.4.5 Let be a graph of size 2. Then ( ( )) = if and only if is a complete graph. 퐺 푚 ≥ 훾0 퐽 퐺 푚 퐺 Proof: Arriola (2015) showed that for any graph of order , ( ) = if and only if is a null graph. Since the jump graph of a complete graph is a null 0 graph, it follows that the isolate domination number of a jump퐺 graph 푛of 훾 is퐺 equal푛 to its size. 퐺 퐺 3. Conclusion∎

In summary, this paper was able to derive and characterized the isolate dominating set of some special graphs such as cocktail party graph, crown graph, gear graph, and jump graph. It was further shown that the following are the exact value of the corresponding isolate domination number:

1. The isolate domination number of any cocktail party graph is 2. 2. The isolate domination number of any crown graph is 2. 3. The isolate domination number of a gear graph with vertices is . 2푛 푛 � 3 � References

Arriola, B. H. (2015). Isolate Domination in the Join and Corona of Graphs, Applied Mathematical Sciences, 9(31), 1543-1549.

Bacak-Turan, G. and Demirtekin, E. (2017). Neighbor Rupture Degree of Gear Graphs. CBU J. of Sci., 12(2), 319-323.

Biggs, N. L. (1993). Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press.

Brandstadt, A., Le, V. and Spinrad, J. (1999). Graph Classes: a survey, SIAM Monographs on Discrete Mathematics and Applications, Springer

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Kirlangic, A. (2009). The Rupture Degree and Gear Graphs. Bulletin of the Malaysian Mathematical Sciences Society, 32(1), 31-36.

Kulli, V. R. and Muddebihal, M. H. (2006). The Lict graph and litact graph of a graph, J. of Analysis and Comput., 2(1), 33-43.

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