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International J.Math. Combin. Vol.3(2017), 106-118 A Study on Equitable Triple Connected Domination Number of a Graph M. Subramanian DepartmentofMathematics AnnaUniversityRegionalCampus-Tirunelveli,Tirunelveli-627007,India T. Subramanian DepartmentofMathematics,UniversityVocCollegeofEngineering AnnaUniversity:ThoothukudiCampus,Thoothukudi-628008,India E-mail: [email protected],[email protected] Abstract: AgraphGissaid tobetriple connected ifanythreeverticeslieonapathinG. A dominating set S of a connected graph G is said to be a triple connected dominating set of G if the induced subgraph S is triple connected. The minimum cardinality taken over h i all tripleconnected dominatingsets iscalled thetriple connected domination number andis denotedbyγtc.AtripleconnecteddominatingsetS ofV inGissaidtobeanequitabletriple connected dominating setifforeveryvertexuinV S thereexistsavertexv inS suchthat − uvisanedgeofGand deg(v) deg(u) 1.Theminimumcardinalitytakenoverallequitable | − |≤ triple connected dominating sets is called the equitable triple connected domination number and is denoted by γetc. In this paper we initiate a study on this parameter. In addition, we discuss the related problem of finding the stability of γetc upon edge addition on some classes of graphs. Key Words: Connected domination, triple connected domination, equitable triple con- necteddominatingset,equitabletripleconnecteddomination number,Smarandachelyequi- table dominating set. AMS(2010): 05C69. §1. Introduction Byagraph,wemeanafinite,simple,connectedandundirectedgraphG(V,E),whereV denotes its vertex set and E its edge set. Unless otherwise stated, the graph G is connected and has p vertices and q edges. For graph theoretic terminology, we refer to Harary [1]. Definition 1.1([2]) A subset S of V in G is called a dominating set of G if every vertex in V S is adjacent toat leastonevertexinS.Thedomination numberγ(G)of Gis theminimum − cardinality taken over all dominating sets in G. 1ReceivedJanuary05,2017,Accepted August25,2017. AStudyonEquitableTripleConnectedDominationNumberofaGraph 107 Definition 1.2([6]) A dominating set S of V in G is said to be an equitable dominating set if for every vertex u in V S there exists a vertex v in S such that uv is an edge of G and − deg(v) deg(u) 1, and Smarandachely equitable dominating set if deg(v) deg(u) 1 for | − |≤ | − |≥ all such an edge. The minimum cardinality taken over all equitable dominating sets in G is the equitable domination number of G and is denoted by γ . e Definition 1.3([2]) A dominating set S of V in G is said to be a connected dominating set of G if the induced sub graph S is connected. The minimum cardinality taken over all connected h i dominating sets in G is the connected domination number of G and is denoted by γ . c Definition1.4([3]) AconnecteddominatingsetS ofV inGissaidtobeanequitableconnected dominating set if for every vertex u in V S there exists a vertex v in S suchthat uv is an edge − of G and deg(v) deg(u) 1. The minimum cardinality taken over all equitable connected | − | ≤ dominating sets in G is the equitable connected domination number of G and is denoted by γ ec. The concept of triple connected graphs has been introduced by Paulraj Joseph et. al. [5] byconsideringtheexistenceofapathcontaininganythreeverticesofG.Theyhavestudiedthe properties of triple connected graphs and established many results on them. A graphG is said tobetriple connectedifanythreeverticeslieonapathinG.Allpaths,cycles,completegraphs and wheels are some standard examples of triple connected graphs. But the star K ,p 4 1,p 1 − ≥ is not a triple connected graph. Definition 1.5([4]) A dominating set S of a connected graph G is said to be a triple connected dominating set of G if the induced sub graph S is triple connected. The minimum cardinality h i taken over all connected dominating sets is the triple connected domination number and is denoted by γ . tc In this paper, we extend the concept of triple connected domination to an equitable triple connected domination and study its properties. Notation 1.6 Let G be a connected graph on m vertices v ,v ,...,v . The graph obtained 1 2 m from G by attaching n times a pendant vertex of P on the vertex v ,n times a pendant 1 l1 1 2 vertexofP onthevertexv andsoon,isdenotedbyG(n P ,n P ,n P , ,n P )where l2 2 1 l1 2 l2 3 l3 ··· m lm n ,l 0 and 1 i m. i i ≥ ≤ ≤ Notation 1.7 We have C (nP ,0,0, ,0)=C (0,nP ,0, ,0)= =C (nP ). p k ··· ∼ p k ··· ∼···∼ p k Example 1.8 The graph G in Figure 1 is isomorphic to C (2P ). 1 3 2 v4 v5 v 3 v v 1 2 G Figure 1 The graph C (2P ). 3 2 108 M.SubramanianandT.Subramanian Proposition 1.9 For any connected graph G with p vertices, 1 γ (G) p. e ≤ ≤ Proposition 1.10 For any connected graph G with p vertices, 1 γ (G) p. ec ≤ ≤ §2. Equitable Triple Connected Domination Number of a Graph In this section, we define the concept of equitable triple connected domination number of a graph. Definition 2.1 A subset S of V of a nontrivial graph G is said to be an equitable triple connected dominating set, if S is triple connected and for every vertex u in V S there exists h i − avertexv in S suchthat uv is anedgeofGand deg(v) deg(u) 1.Theminimumcardinality | − |≤ taken over all equitable triple connected dominating sets is called an equitable triple connected domination number of G and is denoted by γ (G). Any equitable triple connected dominating etc set with γ vertices is called a γ -set of G. etc etc Example 2.2 For the graph G in Figure 2, S = v ,v ,v ,v forms a γ -set of G . Hence 1 2 3 5 6 etc 1 { } γ (G )=4. etc 1 v 5 v 4 v 3 v1 v2 v 6 Figure 2 Graph with γ =4. etc Remark 2.3 Any equitable triple connected dominating set is obviously equitable connected dominatingsetandanyequitable connecteddominating setis alsoanequitable dominatingset and finally and any equitable dominating set is a dominating set. So it is permissible for the equitable triple connected dominating set S can have less than three vertices. If S has 1 (or 2) vertex (vertices) then S can be viewed as an equitable dominating set (or connected equitable dominating set). Throughout this paper, we consider only connected graphs for which equitable triple connected dominating set exists. Definition 2.4 A bistar, denoted by B(m,n) is the graph obtained by joining the centers of the stars K and K . The center of a star K with p > 2 vertices is its unique vertex 1,m 1,n 1,p 1 − of maximum degree. Definition 2.5 A helm graph, denoted by H is the graph obtained from the wheel W by n n attaching a pendant vertex to each vertex in the outer cycle of W (the number of vertices of n H is, p=2n 1). n − AStudyonEquitableTripleConnectedDominationNumberofaGraph 109 Definition 2.6 The friendship graph F is the graph constructed by identifying n copies of the n cycle C at a common vertex. 3 Remark 2.7 It is to be noted that not every graph has a triple connected dominating set likewise not all graphs have an equitable triple connected dominating set. For example, the star graph K does not have an equitable triple connected dominating set. 1,3 §3. Preliminary Results We now proceed to determine the equitable triple connected domination number for some standard graphs. p if p=1 (1) For any path of order p,γetc(Pp)= p 1 if p=2  p−2 if p 3. − ≥ (2) For any cycle of order p,γetc(Cp)=p−2. (3) For any complete bipartite graph other than star of order p=m+n, 2 if m n 1 and m,n=1 γ (K )= | − |≤ 6 etc m,n  p if m n 2 and m,n=1.  | − |≥ 6  (4) For any complete graph of order p,γ (K )=1. etc p 1 if p=4,5 (5) For any wheel of order p,γetc(Wp)= 3 if p=6  p 4 if p 7 − ≥ Equitable triple connected dominating set does not exist for the following special graphs: (6) For any star K other than K . 1,p 1 1,2 − (7) Helm graph H . n (8) Bistar B(m,n). Consider any star K of order p > 3. Let v be its center and v ,v ,..., v be the 1,p 1 1 2 p 1 − − pendant vertices which are adjacent to v. Since every minimum equitable dominating set S must contain all the pendant vertices v ,v , ,v and we have S is not triple connected 1 2 p 1 ··· − h i if p 1 > 2. Hence γ (K ) does not exist if p > 3. Similarly we can prove all the other etc 1,p 1 − − results. Lemma 3.1 If γ (G)=1, then γ (G)=1. e etc Lemma 3.2 If γ (G)=1 (or 2 or 3), then γ (G)=1 (or 2 or 3). ec etc Lemma 3.3 If γ (G)=4, then γ (G) need not be equal to 4. ec etc For C (2P ),γ (C (2P ))=4, but γ (C (2P ))-set does not exist. 3 2 ec 3 2 etc 3 2 110 M.SubramanianandT.Subramanian Theorem 3.4 For any connected graph G with p vertices, we have 1 γ (G) p and the etc ≤ ≤ bounds are sharp. Proof The lower bound follows from Definition 2.1 and the upper bound is obvious. For K the lower bound is attained and for K the upper bound is attained. 4 2,4 2 Observation 3.5 For any connected graph G of order 1, γ (G) = p if and only if G is etc isomorphic to K . 1 Lemma 3.6 There exists no connected graph G with 2 p 4 vertices such that γ (G)=p. etc ≤ ≤ Proof The proof is divided into cases following. Case 1. The only connected graph with of order 2 is K and for K ,γ (K ) = 1 = p 1 2 2 etc 2 − ([1]). Case 2. There are only two connectedgraphs with three verticeswhich are P or K and for 3 3 G=P ,K ,γ (G)=1=p 2 ([1]). ∼ 3 3 etc − Case3. Thevariouspossibilitiesofconnectedgraphsonfourverticesare: K ,P ,C (P ),C ,K 1,3 4 3 2 4 4 − e ,K . If G is isomorphic to P ,C ,C (P ), γ (G) = 2 = p 2. If G is isomorphic to 4 4 4 3 2 etc { } − K ,K e ,γ (G)=1=p 3. If G=K ,γ (G) does not exist. 4 4−{ } etc − ∼ 1,3 etc 2 Theorem 3.7 Let G be a connected graph with p=5 vertices, then γ (G)=p if and only if etc G is isomorphic to C 2K . 3 1 ∪ Proof ([1]) For the various possibilities of connected graphs on five vertices are: K , 1,4 P (0,P ,P ),P , C (2P ), C (P ,P ,0), C (P ), C (P ), C , F , P , K ,K (P ),C 2K , 3 2 2 5 3 2 3 2 2 3 3 4 2 5 2 5 2,3 4 2 3 1 P P ,P 2K ,W ,K e ,K andanyoneofthefollowinggraphsfromG toG inFigure 2 3 3 1 5 5−{ } 5 1 3 S 3. S S v1 v1 v1 v5 v v5 v2 v5 v2 2 v4 v3 v4 v3 v4 v3 G1 G2 G3 Figure 3 Graphs on 5 vertices. IfG=K ,W ,K e ,thenγ (G)=1=p 4.IfG=K (P ), C (P ),K ,P , P 2K , ∼ 5 5 5−{ } etc − ∼ 4 2 3 3 2,3 5 3∪ 1 P P ,G , then γ (G) = 2 = p 3. If G = P ,C ,F , C (P ),G ,G , then γ (G) = 3 = 2∪ 3 3 etc − ∼ 5 5 2 4 2 1 2 etc p 2. If G = C (P ,P ,0), then γ (G) = 4 = p 1. If G = C 2K , then γ (G) = 5 = p. − ∼ 3 2 2 etc − ∼ 3∪ 1 etc If G=K ,C (2P ), P (0,P ,P ), then γ (G) does not exist. ∼ 1,4 3 2 3 2 2 etc 2 Theorem 3.8 Let G be a connected graph with p=6 vertices, then γ (G)=p if and only if etc AStudyonEquitableTripleConnectedDominationNumberofaGraph 111 G is isomorphic to K or any one of the graphs: G ,G ,G in Figure 4. 2,4 1 2 3 v1 v2 v1 v2 v v 1 2 v3 v4 v3 v4 v 3 v 4 v5 v6 v5 v6 v v 5 6 G1 G2 G3 Figure 4 Graphs on 6 vertices with γ (G)=6. etc Proof Let G be a connected graph with p = 6 vertices, and let γ (G) = 6 ([1]). Among etc all of the connected graphs on 6 vertices, it can be easily verified that G=K or any one of ∼ 2,4 the graphs G ,G ,G in Figure 4.The converse part is obvious. 1 2 3 2 Lemma 3.9 Let G be a connected graph of order 2 such that γ(G)=γ (G). Then G=K . etc ∼ 2 Lemma3.10 LetGbeaconnectedgraphoforder3suchthatγ(G)=γ (G).ThenG=K ,P . etc ∼ 3 3 Lemma 3.11 Let G be a connected graph of order 4 such that γ(G) = γ (G). Then G is etc isomorphic to one of the following graphs: P ,C ,K ,K e . 4 4 4 4 −{ } Proof Forthevariouspossibilitiesofconnectedgraphsonfourverticesare: K ,P ,C (P ), 1,3 4 3 2 C ,K e ,K . If G=P ,C ,γ(G)=γ (G)=2. If G=K ,K e ,γ(G)=γ (G)=1. 4 4−{ } 4 ∼ 4 4 etc ∼ 4 4−{ } etc If G = C (P ),K ,γ(C (P )) = 1 but γ (C (P )) = 2 and γ(K ) = 1 but γ (K ) does ∼ 3 2 1,3 3 2 etc 3 2 1,3 etc 1,3 not exist. Hence the lemma. 2 Theorem 3.12 Let G be a connected graph on order 5 such that γ(G) = γ (G). Then G is etc isomorphic to one of the following graphs: C (P ),P , K ,P P ,W ,K e ,K . 3 3 5 2,3 2 3 5 5 5 ∪ −{ } Proof Forthevariouspossibilitiesofconnectedgraphsonfiveverticesare: K ,P (0,P ,P ), 1,4 3 2 2 P ,C (2P ),C (P ,P ,0),C (P ),C (P ),C ,F ,P ,K ,K (P ), C 2K , P P ,P 2K , 5 3 2 3 2 2 3 3 4 2 5 2 5 2,3 4 2 3 1 2 3 3 1 ∪ ∪ ∪ W ,K e ,K andanyoneofthefollowinggraphsfromG toG inFigure3. Amongallthe 5 5 5 1 3 −{ } above possibilities it can be easily verified that γ(G) = γ (G) only if G = C (P ),P ,K , etc ∼ 3 3 5 2,3 P P , W ,K e ,K . 2 3 5 5 5 ∪ −{ } 2 Theorem 3.13 LetG be aconnected graph of order 6such that γ(G)=γ (G). Then G is iso- etc morphictoK ,K e ,K ,oranyoneofthegraphs: G ,G ,G ,G ,G ,G ,G ,G ,G ,G , 3,3 6 6 1 2 3 4 5 6 7 8 9 10 −{ } G ,G in Figure 5. 11 12 112 M.SubramanianandT.Subramanian v1 v2 v1 v2 v1 v2 v v3 v4 3 v4 v3 v 4 v5 v6 v5 v6 v5 v6 G1 G2 G3 v1 v2 v1 v2 v1 v2 v3 v4 v3 v4 v3 v4 v5 v6 v5 v6 v5 v6 G G5 G6 4 v1 v2 v1 v2 v1 v2 v3 v4 v3 v4 v3 v4 v5 v6 v5 v6 v5 v6 G G8 G9 7 G10 G11 G12 Figure 5 Graphs on 6 vertices such that γ(G)=γ (G). etc Proof Let G be a connected graph of order 6 such that γ(G) = γ (G). It is straight etc forwardtoobservethatγ(G)=γ (G)onlyifG=K ,K e ,K oranyoneofthe graphs etc ∼ 3,3 6−{ } 6 G ,G ,G ,G ,G ,G ,G ,G ,G ,G ,G ,G in Figure 5. 1 2 3 4 5 6 7 8 9 10 11 12 2 Observation 3.14 For any connected graphG,γ (G) γ (G) γ (G) and the bounds can e ec etc ≤ ≤ be strict as well as sharp for all possible cases. (1) For the complete graph K ,γ (K )=γ (K )=γ (K )=1. 5 e 5 ec 5 etc 5 (2) For K (P ),γ (K (P ))=2<γ (K (P ))=γ (K (P ))=3. 4 3 e 4 3 ec 4 3 etc 4 3 (3) For the graph G in Figure 6, γ (G )=3<γ (G )=4<γ (G )=5. 1 e 1 ec 1 etc 1 (4) For the graph G in Figure 6, γ (G )=γ (G )=4<γ (G )=5. 2 e 2 ec 2 etc 2 AStudyonEquitableTripleConnectedDominationNumberofaGraph 113 v1 v2 v1 v2 v3 v4 v3 v4 v5 v6 v5 v6 G G 2 1 Figure 6 Lemma 3.15 Let G be a connected graph of order 1 such that γ (G)=γ (G)=γ (G). Then e ec etc G=K . ∼ 1 Lemma 3.16 Let G be a connected graph of order 2 such that γ (G)=γ (G)=γ (G). Then e ec etc G=K . ∼ 2 Lemma 3.17 Let G be a connected graph of order 3 such that γ (G)=γ (G)=γ (G). Then e ec etc G=K ,P . ∼ 3 3 Lemma 3.18 Let G be a connected graph of order 4 such that γ (G)=γ (G)=γ (G). Then e ec etc G is isomorphic to one of the following graphs: P ,C ,K ,C (P ),K e . 4 4 4 3 2 4 −{ } Proof The various possibilities of connected graphs on four vertices are: K ,P ,C (P ), 1,3 4 3 2 C ,K e ,K . If G = P ,C ,C (P ),γ (G) = γ (G) = γ (G) = 2. If G = K ,K e , 4 4−{ } 4 ∼ 4 4 3 2 e ec etc ∼ 4 4−{ } γ (G) = γ (G) = γ (G) = 1. And if G = K ,γ (G) = γ (G) = 4 and γ (G) does not e ec etc ∼ 1,3 e ec etc exist. Hence the lemma. 2 Theorem 3.19 Let G be a connected graph of order 5 such that γ (G) = γ (G) = γ (G). e ec etc Then G is isomorphic to one of the following graphs: C (P ),C (P ),P ,K ,F ,K (P ), 3 3 4 2 5 2,3 2 4 2 P P ,P 2K ,W ,K e ,K or the graphs: G to G in Figure 3. 2 3 3 1 5 5 5 1 3 ∪ ∪ −{ } Proof Forthevariouspossibilitiesofconnectedgraphsonfiveverticesare: K ,P (0,P ,P ), 1,4 3 2 2 P ,C (2P ),C (P ,P ,0),C (P ),C (P ),C ,F ,P ,K ,K (P ),C 2K ,P P ,P 2K , 5 3 2 3 2 2 3 3 4 2 5 2 5 2,3 4 2 3 1 2 3 3 1 ∪ ∪ ∪ W ,K e ,K and any one of the following graphs from G to G in Figure 3. If G = 5 5 −{ } 5 1 3 ∼ K ,W ,K e ,thenγ (G)=γ (G)=γ (G)=1.IfG=K (P ),C (P ),K ,P ,P 2K , 5 5 5−{ } e ec etc ∼ 4 2 3 3 2,3 5 3∪ 1 P P ,G , then γ (G) = γ (G) = γ (G) = 2. If G = F or C (P ) then γ (G) = γ (G) = 2∪ 3 3 e ec etc ∼ 2 4 2 e ec γ (G) =3. If G=G or G then γ (G) =2, but γ (G)= γ (G) =3. If G= C (P ,P ,0), etc ∼ 1 2 e ec etc ∼ 3 2 2 then γ (G) = 3, but γ (G) = γ (G) = 4. If G = C 2K , thenγ (G) = γ (G) = 4, e ec etc ∼ 3∪ 1 e ec but γ (G) = 5. If G = K , then γ (G) = γ (G) = 5, but γ (G) does not exist. If etc ∼ 1,4 e ec etc G = P (0,P ,P ), then γ (G) = 3,γ (G) = 4 and γ (G) does not exist. If G = C (2P ) ∼ 3 2 2 e ec etc ∼ 3 2 then γ (G) = γ (G) = 4, but γ (G) does not exist. If G = P ,C , then γ (G) = 2,but e ec etc ∼ 5 5 e γ (G)=γ (G)=3. ec etc 2 114 M.SubramanianandT.Subramanian Theorem 3.20 Let G be a connected graph of order 6 such that γ (G) = γ (G) = γ (G). e ec etc Then G=K . ∼ 2,4 Proof Let G be a connectedgraphof order6 such that γ (G)=γ (G)=γ (G). Among e ec etc all of the connected graphs on 6 vertices,it can be easily seen that K is the only graph with 2,4 γ (G)=γ (G)=γ (G)=6. e ec etc 2 Theorem 3.21 If G is a connected graph of order p = 2n for some positive integer n 2 ≥ such that its vertex set and edge set are V(G) = v : 1 i p and E(G) = v v : 1 i i i+1 { ≤ ≤ } { ≤ i p 1 v v : for i = 1 to p and j = P +1 to p respectively, then γ (G) = γ (G) = ≤ − }∪{ i j 2 2 } e ec γ (G)=n 1. etc − (cid:0) (cid:1) Example 3.22 For p= 6(=2n), By Theorem 3.21, the graph constructed is shown in Figure 7. Clearly any two adjacent vertices from the set v ,v ,v ,v forms a minimum equitable 2 3 4 5 { } triple connected dominating set. Hence γ (G)=2=n 1. etc − v1 v2 v3 v4 v5 v6 G 1 Figure 7 Graph illustrating the Theorem 3.21 Proposition 3.23 Let G be a triple connected graph on order p. If its vertex set V(G) can be partitioned into k sets S ,S , ,S such that S = v : deg(v) = m , S = v : deg(v) = 1 2 k 1 1 2 { ··· } { } { m m +2 , S = v : deg(v) = m m +2 , ,S = v : deg(v) = m m +2 2 1 3 3 2 k k k 1 ≥ } { ≥ } ··· { ≥ − } where m ’s are increasing sequence of positive integers and S is K for some positive integer i i n h i n, for 1 i k, then γ (G)=γ (G)=γ (G)=p. e ce etc ≤ ≤ Remark 3.24 The converseof Proposition3.23 need not be true. Let G be a triple connected graph given in Figure 8. Clearly γ (G)=γ (G) =γ (G)=p, but there is no such partition e ec etc of V(G) as stated in Proposition 3.23. Since V(G) can be partitioned in to S = v ,v of 1 11 12 { } degree 1,S = v ,v ,v ,v ,v ,v of degree 2,S = v of degree 3,S = v ,v of degree 2 3 4 5 7 8 9 3 2 4 6 10 { } { } { } 4 and S = v of degree 7 such that S is totally disconnected, for 1 i 5. 5 1 i { } h i ≤ ≤ AStudyonEquitableTripleConnectedDominationNumberofaGraph 115 v 1 v2 v3 v4 v5 v7 v8 v9 v 10 v11 v6 v 12 G 1 Figure 8 Counter example for Proposition3.23 Lemma 3.25 Let T be any tree, γ (T)=p if and only if T =K . etc ∼ 1 Proof Let T = K , then clearly γ (T) = p. Conversely, let T be a tree such that ∼ 1 etc γ (T) = p. Now T is triple connected, it follows that T = P ([5] since a tree T is triple etc h i ∼ p connected if and only if T =P ;p 3) and given that γ (T)=p, we have T =K . ∼ p ≥ etc ∼ 1 2 Lemma 3.25 Let T be any tree, γ (T)=p 1 if and only if T =K . etc − ∼ 2 Proof Let T = K , then clearly γ (T) = p 1. Conversely, let T be a tree such that ∼ 2 etc − γ (T)=p 1.Let v be the vertexnot in γ (T)-set. Suppose deg(v )>1, then we can find etc p etc p − a cycle in T,which is a contradiction. Hence deg(v )=1.Since v is a pendant vertexwe have p p T v isalsoatree. Then T v istripleconnected,whichfollowsthatT v =P −{ p} h −{ p}i −{ p}∼ p−1 (from [5]) and hence T =P and given that γ (T)=p 1, we have T =K . ∼ p etc − ∼ 2 2 Theorem 3.27 Let T be any tree on p>2 vertices. Then either γ (T)=p 2 if T =P or etc − ∼ p γ -set does not exist. etc Proof The proof is divided into cases following. Case 1. If T contains two pendant vertices. Then T = P for which γ (T) = p 2, where ∼ p etc − p>2. Case 2. If T contains more than two pendant vertices. Since any equitable triple connected dominating set must contain all the pendant vertices or its supports and also T is connected and acyclic it follows that γ -set does not exist. etc 2 §4. Equitable Triple Connected Domination Edge Addition Stable Graphs In this section, we consider the problem of finding the stability of γ upon edge addition of etc some classes of graphs such as cycles and complete bipartite graphs.