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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 Arriola J-HERD Vol.2. Issue 2. 2017 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 illustrates퐿��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 〉 24 Arriola J-HERD Vol.2. Issue 2. 2017 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 an퐿� 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 25 Arriola J-HERD Vol.2. Issue 2. 2017 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 26 Arriola J-HERD Vol.2. Issue 2. 2017 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 ∅ 27 Arriola J-HERD Vol.2. Issue 2. 2017 ii. = such that , ( ) ( ) = ( ) , and either there exists with ( ) = or 푆 푆1 ∪ 푆2 푆1 ≠ ∅ 푁퐺푛 푆1 ∪ 푁퐺푛 푆2 푉 퐺푛 there exists with ( ) = , 1 퐺푛 2 .
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