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# Characterization of trees with equal total edge domination and double edge domination numbers

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### Characterization of trees with equal total edge domination and double edge domination numbers

1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012 Characterization of Trees with Equal Total Edge Domination and Double Edge Domination Numbers M.H.Muddebihal A.R.Sedamkar* Department of Mathematics, Gulbarga University, Gulbarga-585106, Karnataka, INDIA E-mail of the corresponding author: mhmuddebihal@yahoo.co.inAbstractA total edge dominating set of a graph G is a set D of edges of G such that the sub graph D has no isolatededges. The total edge domination number of G denoted by γ t (G ) , is the minimum cardinality of a total edge dominating set of G . Further, the set D is said to be double edge dominating set of graph G . If every edge of G isdominated by at least two edges of D . The double edge domination number of G , denoted by, γ d (G ) , is theminimum cardinality of a double edge dominating set of G . In this paper, we provide a constructivecharacterization of trees with equal total edge domination and double edge domination numbers.Key words: Trees, Total edge domination number, Double edge domination number. 1. Introduction: In this paper, we follow the notations of [2]. All the graphs considered here are simple, finite, non-trivial,undirected and connected graph. As usual p = V and q = E denote the number of vertices and edges of agraph G , respectively. In general, we use X to denote the sub graph induced by the set of vertices X and N ( v ) andN [ v ] denote the open and closed neighborhoods of a vertex v , respectively. The degree of an edge e = uv of G is defined by deg e = deg u + deg v − 2 , is the number of edgesadjacent to it. An edge e of degree one is called end edge and neighbor is called support edge of G . A strong support edge is adjacent to at least two end edges. A star is a tree with exactly one vertex of degreegreater than one. A double star is a tree with exactly one support edge. 72
2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012 For an edge e is a rooted tree T , let C ( e ) and S ( e ) denote the set of childrens and descendants of erespectively. Further we define S [e] = S (e) U {e} . The maximal sub tree at e is the sub tree of T inducedby S [e] , and is denoted by Te . A set D ⊆ E is said to be total edge dominating set of G , if the sub graph D has no isolated edges. Thetotal edge domination number of G , denoted by γ t (G ) , is the minimum cardinality of a total edge dominating setof G . Total edge domination in graphs was introduced by S.Arumugam and S.Velammal [1]. A set S ⊆ V is said to be double dominating set of G , if every vertex of G is dominated by at least twovertices of S . The double domination number of G , denoted by γ d (G ) , is the minimum cardinality of a doubledominating set of G . Double domination is a graph was introduced by F. Harary and T. W. Haynes [3]. The conceptof domination parameters is now well studied in graph theory (see [4] and [5]). Analogously, a set D ⊆ E is said to be double edge dominating set of G , if every edge of G is dominatedby at least two edges of D . The double edge domination number of G , denoted by γ d (G ) , is the minimumcardinality of a double edge dominating set of G . In this paper, we provide a constructive characterization of trees with equal total edge domination anddouble edge domination numbers. 2. Results:Initially we obtain the following Observations which are straight forward.Observation 2.1: Every support edge of a graph G is in every γ t (G ) set.Observation 2.2: Suppose every non end edge is adjacent to exactly two end edge, then every end edge of a graphG is in every γ d (G ) set. Observation 2.3: Suppose the support edges of a graph G are at distance at least three in G , then every supportedge of a graph G is in every γ d (G ) . 73
3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012 3. Main Results:Theorem 3.1: For any tree T , γ d (T ) ≥ γ t (T ) . Proof: Let q be the number of edges in tree T . We proceed by induction on q . If diam (T ) ≤ 3 . Then T is eithera star or a double star and γ d (T ) = 2 = γ t (T ) . Now assume that diam (T ) ≥ 4 and the Theorem is true forevery tree T with q < q . First assume that some support edge of T , say ex is strong. Let ey and ez be the endedges adjacent to ex and T = T − e . Let D be any γ d (T ) - set. Clearly ex ∈ D , where D is a total edge dominating set of tree T . Therefore γ t (T ) ≤ γ t (T ) . Now let S be any γ d (T ) - set. By observations 2 and 3, wehave e y , e x , e z ∈ S . Clearly, S − {ey } is a double edge dominating set of tree T . Thereforeγ d (T ) ≤ γ d (T ) − 1. Clearly, γ d (T ) ≥ γ d (T ) + 1 ≥ γ t (T ) + 1 ≥ γ t (T ) + 1 ≥ γ t (T ) , a contradiction.Therefore every support edge of T is weak. Let T be a rooted tree at vertex r which is incident with edge er of the diam (T ) . Let et be the endedge at maximum distance from er , e be parent of et , eu be the parent of e and ew be the parent of eu in therooted tree. Let Tex denotes the sub tree induced by an edge ex and its descendents in the rooted tree. Assume that degT (eu ) ≥ 3 and eu is adjacent to an end edge ex . Let T = T − e and D be theγ t (T ) - set. By Observation 1, we have eu ∈ D . Clearly, D U {e} is a total edge dominating set of tree T . Thusγ t (T ) ≤ γ t (T ) + 1 . Now let S be any γ d (T ) - set. By Observations 2 and 3, et , e x , e, eu ∈ S . Clearly,S − {e, et } is a double edge dominating set of tree T . Therefore γ d (T ) ≤ γ d (T ) − 2 . It follows thatγ d (T ) ≥ γ d (T ) + 2 ≥ γ t (T ) + 2 ≥ γ t (T ) +1 ≥ γ t (T ) . 74
4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012 Now assume that among the decedents of eu there is a support edge say ex , which is different from e . LetT = T − e and D be the γ t (T ) - set containing no end edges. Since ex must have a neighbor in D , thuseu ∈ D . Clearly D U {e} is a total edge dominating set of tree T and hence γ t (T ) ≤ γ t (T ) + 1 . Now let S beany γ d (T ) - set. By Observations 2 and 3, we have et , e, e x ∈ S . If eu ∈ S , then S − {e, et } is the double edge dominating set of tree T . Further assume that eu ∉ S . Then S U {eu } − {e, et } is a double edge dominating set of tree T . Therefore γ d (T ) ≤ γ d (T ) − 1 . Clearly, it follows that, γ d (T ) ≥ γ d (T ) + 1 ≥ γ t (T ) + 1 ≥ γ t (T ) . Therefore, we obtain γ d (T ) ≥ γ t (T ) . To obtain the characterization, we introduce six types of operations that we use to construct trees with equal totaledge domination and double edge domination numbers.Type 1: Attach a path P to two vertices u and w which are incident with eu and ew respectively of T where 1eu , ew located at the component of T − ex ey such that either ex is in γ d set of T or ey is in γ d - set of T . Type 2: Attach a path P2 to a vertex v incident with e of tree T , where e is an edge such that T − e has acomponent P . 3Type 3: Attach k ≥ 1 number of paths P3 to the vertex v which is incident with an edge e of T where e is anedge such that either T − e has a component P2 or T − e has two components P2 and P4 , and one end of P4 isadjacent to e is T .Type 4: Attach a path P3 to a vertex v which is incident with e of tree T by joining its support vertex to v , suchthat e is not contained is any γ t - set of T .Type 5: Attach a path P4 ( n ) , n ≥ 1 to a vertex v which is incident with an edge e , where e is in a γ d - set of T if n = 1 . 75
5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012Type 6: Attach a path P5 to a vertex v incident with e of tree T by joining one of its support to v such thatT − e has a component H ∈{P3 , P4 , P6 }.Now we define the following families of treesLet ℑ be the family of trees with equal total edge domination number and double edge domination number. That isℑ= {T / T is a tree satisfying γ t (T ) = γ d ( T )} . We also define one more family asℜ = {T / T is obtained from P3 by a finite sequence of type - i operations where 1 ≤ i ≤ 5} .We prove the following Lemmas to provide our main characterization.Lemma 3.2: If T ∈ ℑ and T is obtained from T by Type-1 operation, then T ∈ ℑ . Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t (T ) ≤ γ d (T ) , we only need to prove that γ t (T ) ≥ γ d (T ) . Assume that T is obtained from T by attaching the path P to two vertices u and w which are 1incident with eu and ew as eu and ew respectively. Then there is a path ex ey in T such that either ex is in γ d - set of T and T − ex ey has a component P5 = eu eew ex or ey is in γ d - set of T and T − ex ey has a component P6 = eu eew ex ex . Clearly, γ t (T ) = γ t (T ) − 1 .Suppose T − ex ey contains a path P5 = eu eew ex then S be the γ d - set of T containing ex . From Observation 2 and by the definition of γd - set, we have S I {eu , e , ew , ex } = {eu , ew } or {eu , e} . Therefore 76
6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012 S = ( S − {eu , e , ew }) U {eu , e, ew } is a double edge dominating set of T with S = S + 1 = γ d (T ) + 1 . Clearly, γ t (T ) = γ t (T ) + 1 = γ d (T ) + 1 = S > γ d (T ) . Now, if T − ex ey contains a path P6 = eu e ew ex ex . Then S be the γ d - set of T containing ey . By Observation 2 and by definition of γd - set, we have S I { eu , e, ew , ex , ex } = { eu , ew , ex } . Therefore S = ( S − {eu , ew }) U {eu , e, ew } is a double edge dominating set of T with S = S + 1 = γ d (T ) + 1 . Clearly, γ t (T ) = γ t (T ) + 1 = γ d (T ) +1 = S ≥ γ d (T ) . Lemma 3.3: If T ∈ ℑ and T is obtained from T by Type-2 operation, then T ∈ ℑ .Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t (T ) ≤ γ d (T ) , we only need to prove thatγ t (T ) ≥ γ d (T ) . Assume that T is obtained from T by attaching a path P2 to a vertex v which is incident withe of T where T − e has a component P3 = ewex . We can easily show that γ t (T ) = γ t (T ) + 1 . Now bydefinition of γd - set, there exists a γd - set, D of T containing the edge e . Then D U {eu } forms a double edge dominating set of T . Therefore γ t (T ) = γ t (T ) + 1 = γ d (T ) + 1 = D U {eu } ≥ γ d (T ) . Lemma 3.4: If T ∈ ℑ and T is obtained from T by Type - 3 operation, then T ∈ ℑ . Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t (T ) ≤ γ d (T ) , hence we only need to prove that γ t (T ) ≥ γ d (T ) . Assume that T is obtained from T by attaching m ≥ 1 number of paths P3 to a vertex vwhich is incident with an edge e of T such that T − e has a component P3 or two components P2 and P4 . Bydefinition of γ t - set and γd - set, we can easily show that γ t (T ) ≥ γ t (T ) + 2m and γ d (T ) + 2m ≥ γ d (T ) . Since γ t (T ) = γ d (T ) , it follows that γ t (T ) ≥ γ t (T ) + 2m = γ d (T ) + 2m ≥ γ d (T ) . 77
7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012Lemma 3.5: If T ∈ ℑ and T is obtained from T by Type - 4 operation, then T ∈ ℑ .Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t (T ) ≤ γ d (T ) , hence we only need to prove that γ t (T ) ≥ γ d (T ) . Assume that T is obtained from T by attaching path P3 to a vertex v incident with e inT such that e is not contained in any γ t -set of T and T − e has a component P4 . For any γ d - set, S of T , S U {ex , ez } is a double edge dominating set of T . Hence γ d (T ) + 2 ≥ γ d (T ) . Let D be any γ t - set of T containing the edge eu , which implies ey ∈ D and D I {e, ex , ez } =1 . If e ∉ D , then D I E (T ) = D − 2 = γ t (T ) − 2 ≥ γ t (T ) , since D I E (T ) is a total edge T . since γ t (T ) = γ d (T ) , that γ t (T ) ≥ γ t (T ) + 2 dominating set of Further it follows = γ d (T ) + 2 ≥ γ d (T ) . If e∈D , then D I {e, ex , ez } = {e} and D I E (T ) = D −1 = γ t (T ) −1 ≥ γ t (T ) , sinceD I E (T ) is a total edge dominating set of T . Suppose γ t (T ) ≤ γ d (T ) − 1 , then by γ d (T ) = γ t (T ) , it follows that γ d (T ) ≥ γ t (T ) + 1≥ γ t (T ) + 2 = γ d (T ) + 2 ≥ γ d (T ) . Clearly, D I E (T ) = γ t (T ) − 1 = γ t (T ) and D I E (T ) is a total edge dominating set of T containing e . Therefore, it gives a contradiction to the fact that e is not in any total edge dominating set of T andhence γ t (T ) ≥ γ d (T ) . Lemma 3.6: If T ∈ ℑ and T is obtained from T by Type - 5 operation, then T ∈ ℑ . Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t (T ) ≤ γ d (T ) , hence we only need to prove that γ t (T ) ≥ γ d (T ) . Assume that T is obtained from T by attaching path P4 ( n ) , n ≥ 1 to a vertex v incident 78
8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012with e in T such that e is in γ d - set for n = 1 . Clearly, γ t (T ) ≥ γ t (T ) + 2 n . If n ≥ 2 , then by γ t (T ) = γ d (T ) , it is obvious that γ t (T ) ≥ γ t (T ) + 2n = γ d (T ) + 2 n ≥ γ d (T ) . If n = 1 , then D be aγd - set of T containing e . Thus D U {ez , e x } is a double edge dominating set of T . Henceγ t (T ) ≥ γ t (T ) + 2 = γ d (T ) +2 = S U {ez , ex } ≥ γ d (T ). Lemma 3.7: If T ∈ ℑ and T is obtained from T by Type - 6 operation, then T ∈ ℑ . Proof: Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . By Theorem 1, γ t (T ) ≤ γ d (T ) , hence we only need to prove that γ t (T ) ≥ γ d (T ) . Assume that T is obtained from T by attaching path a path P5 to a vertex v which isincident with e . Then there exists a subset D of E (T ) as γ t - set of T such that D I NT (e) ≠ φ for n = 1 .Therefore D I E (T ) is a total edge dominating set of T and D I E (T ) ≥ γ t (T ) . By the definition of doubleedge dominating set, we have γ d (T ) + 3 ≥ γ d (T ) . Clearly, it follows thatγ t (T ) = D = D I E ( P6 ) + D I E (T ) > 3 + γ t (T ) = 3 + γ d (T ) ≥ γ d (T ) . Now we define one more family asLet T be the rooted tree. For any edge e ∈ E (T ) , let C ( e ) and F ( e ) denote the set of children edges anddescendent edges of e respectively. Now we define C (e) = {eu ∈ C (e) every edge of F [ eu ] has a distance at most two from e in T } . Further partition C (e) into C1 (e) , C2 (e) and C3 (e) such that every edge of Ci (e) has edgedegree i in T , i = 1, 2 and 3 . 79
9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012Lemma 3.8: Let T be a rooted tree satisfying γ t (T ) = γ d (T ) and ew ∈ E (T ) . We have the followingconditions: If C (ew ) ≠ φ , then C1 (ew ) = C3 (ew ) = φ . 1. If C3 (ew ) ≠ φ , then C1 (ew ) = C2 (ew ) = φ and C3 (ew ) = 1 . 2. 3. If C (ew ) = C (ew ) ≠ C1 (ew ) , then C1 (ew ) = C3 (ew ) = φ .Proof: Let C1 (ew ) = {ex1 , ex2 ,...........exl } , C2 (ew ) = {ey1 , ey2 ,...........eym } and C3 (ew ) ={ez1 , ez2 ,...........ezn } such that C1 (ew ) = l , C2 (ew ) = m and C3 (ew ) = n . For every i = 1, 2,..., n , leteu be the end edge adjacent to e z in T and T = T − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } . i iFor (1): We prove that if m ≥ 1 , then l + n = 0 . Assume l + n ≥ 1 . Since m ≥ 1 , we can have a γd - set S of Tsuch that ew ∈ S and a γ t - set D of T such that ew ∈D . Clearly S − {ex1 , ex2 ,..., exl , eu1 , eu2 , ..., eun } is a double edge dominating set of T and D is a total edge dominating set of T . Henceγ t (T ) = D ≥ γ t (T ) = γ d (T ) = S > S − {ex , ex ,..., ex , eu , eu , …, eu } ≥ γ d (T ) , it gives a contradiction 1 2 l 1 2 nwith Theorem 1.For (2) and (3): Either if C3 (ew ) ≠ φ or if C (ew ) = C (ew ) ≠ C1 (ew ) . Then for both cases, m + n ≥ 1 . Now T such that ew ∈ D . Then D is also a total edge dominating set of T . Hence select a γ t - set D of γ t (T ) = D ≥ γ t (T ) . Further by definition of γd - set and by Observation 2, there exists a γd -set S of Twhich satisfies S I {ey1 , ey2 ,..., eym , ez1 , ez2 ,..., ezn } = φ . Then ( S I E (T )) U {ew } is a γ d -set of T . Hence γ d (T ) ≤ ( S I E (T )) U {ew } ≤ S − (l + n) + 1 = γ d (T ) − (l + n) + 1 = γ t (T ) − (l + n) + 1 . If n ≥ 1 , then γ d (T ) ≤ γ t (T ) ≤ γ t (T ) ≤ γ d (T ) , the last inequality is by Theorem 1. It follows that l + n = 1 and ew ∉ S . Therefore l = 0 and n = 1 . From Condition 1, we have m = 0 . Hence 2 follows. 80
10. 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012If C (ew ) = C (ew ) ≠ C1 (ew ) , then m + n ≥ 1 . By conditions 1 and 2, l = 0 . Now we show that n = 0 . Otherwise, similar to the proof of 2, we have ew ∉ S , n = 1 and m = 0 . Since C (ew ) = C (ew )and deg (ew ) = 2 , for double edge domination, ew, ez ∈S , a contradiction to the selection of S .Lemma 3.9: If T ∈ ℑ with at least three edges, then T∈ℜ.Proof: Let q= E(T) . Since T ∈ ℑ , we have γ t (T ) = γ d (T ) . If diam (T ) ≤ 3 , then T is either a star or a double star and γ t (T ) = 2 = γ d (T ) . Therefore T ∈ ℜ . If diam (T ) ≥ 4 , assume that the result is true for all T with E(T ) = q < q . treesWe prove the following Claim to prove above Lemma.Claim 3.9.1: If there is an edge ea ∈E(T) such that T − ea contains at least two components P3 , then T ∈ ℜ . = eb eb and ec ec are two components of T − ea . If T =T −{eb , eb}, then use D and S to Proof: Assume that P3denote the γ t -set of T containing ea and γ d - set of T , respectively. Since ea ∈ D , D U {eb } is a total edge dominating set of T and hence γ t (T ) ≥ γ t (T) −1. Further since S is a γd - set of T , S I {ea , eb , eb } = {ea , eb } by the definition of γd - set. Clearly, S I E (T ) is a double edge dominating set of T and hence γ t (T ) ≥γ t (T) −1=γ d (T) −1= S I E(T ) ≥ γ d (T ) . By using Theorem 1, we get γ t (T ) = γ d (T ) and so T ∈ ℑ .By induction on T , T ∈ℜ . Now, since T is obtained from T by type - 2 operation, T ∈ ℜ .By above claim, we only need to consider the case that, for the edge ea , T − ea has exactly one component P3 . LetP = eu e ew ex ey ez ...er be a longest path in T having root at vertex r which is incident with er .Clearly, C(ew) =C (ew) ≠ C1 (ew ) .By 3 of Lemma 6, C1 (ew ) = C3 (ew ) = φ . Hence P4 = eu e ew is acomponent of T − ex . Let n be the number of components of P4 of S ( ex ) in T such that an end edge of every 81
11. 11. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012P4 is adjacent to ex . Suppose S ( ex ) in T has a component P4 with its support edge is adjacent to ex . Then itconsists of j number of P3 and k number of P2 components. By Lemma 6, m, j ∈ {0,1} and k ∈ {0,1, 2} .Denoting the n components P4 of the sub graph S ( ex ) in T with one of its end edges is adjacent to an edgeex in T by P4 = eui ei ewi , 1 ≤ i ≤ n . We prove that result according to the values of {m, j , k} .Case 1: Suppose m = j = k =0. Then S ( ex ) = P4 ( n ) , n ≥ 1 in T. Further assume thatT = T − S [ ex ] , then 2 ≤ E (T ) < q . Clearly, γ t (T ) ≥ γ t (T ) − 2n . Let S be a γ d - set of T such that S contains as minimum number of edges of the sub graph S ( ex ) as possible. Then ex ∉ S and S I S[ex ] = 2nby the definition of γd - set. Clearly S I E (T ) is a double edge dominating set of T andhence γ t (T ) ≥ γ t (T ) − 2n = γ d (T ) −2n = S I E (T ) ≥ γ d (T ) . By Theorem 1, γ t (T ) = γ d (T ) and S I E (T ) is a double edge dominating set of T . Hence T ∈ ℑ . By applying the inductive hypothesis to T ,T ∈ℜ .If n ≥ 2 , then it is obvious that T is obtained from T by type - 5 operation and hence T ∈ ℜ .If n = 1 . Then S ( ex ) = P4 = eu e ew in T which is incident with x of an edge ex andS I{eu , e, ew, ex} = {eu , ew} . To double edge dominate, ex , ey ∈ S and so ey ∈ S I E (T ) , which impliesthat ey is in some γ d - set of T . Hence T is obtained from T by type-5 operation and T ∈ℜ . Case 2: Suppose m ≠ 0 and by the proof of Lemma 6, m = 1 and j = k = 0 . Denote the component P4 of S ( ex ) in T whose support edge is adjacent to ex in T by P4 = ea eb ec and if T = T − {ea , eb , ec } . Then,clearly 3 ≤ E (T ) ≤ q . Let S be a γ d - set of T which does not contain eb . 82
12. 12. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012Now we claim that ex is not in any γ t - set of T . Suppose that T has a γ t - set containing ex which is denoted by D , then D U {eb } is a total edge dominating set of T . Clearly, γ t (T ) ≥ γ t (T ) −1 . Since eb ∉ S thenS I E (T ) is a double edge dominating set of T . Henceγ t (T ) ≥ γ t (T ) −1 = γ d (T ) −1 = S I E (T ) + 1≥ γ d (T ) + 1 , which gives a contradiction to the fact thatγ t (T ) ≤ γ d (T ) . This holds the claim and therefore T can be obtained from T by type-4 operation.Now we prove that T ∈ℜ . Let D be any γ t - set of T . By above claim, ex ∉ D . Since D U {ex , eb } is a totaledge dominating set of T , γ t (T ) ≥ γ (T ) − 2 . Further since e t b ∉ S , S I E (T ) is a double edge dominatingset of T , γ t (T ) ≥ γ t (T ) − 2 = γ d (T ) − 2 = S I E (T ) ≥ γ d (T ) . Therefore by Theorem 1, we getγ t (T ) = γ d (T ) , which implies that T ∈ ℑ . Applying the inductive hypothesis on T , T ∈ℜ and henceT ∈ℜ .Case 3: Suppose j ≠0 and by the proof of Lemma 6, m = k = 0 . Let i { T =T − Un=1 eui ,ei ,ewi } . Clearly,3 ≤ E (T ) < q and T is obtained from T by type - 3 operation. We only need to prove that T ∈ℜ . Suppose D ⊂ E (T ) be a γ t - set of T . Then i ( { }) D U Un=1 ei ,ewi is a total edge dominating set of T and hence. γ t (T ) ≥ γ t (T ) − 2n . Since T − ex has acomponent P = ea eb , we can choose 3 S⊆ E(T) as a γd - set of T containing ex . Then S I E(T ) is a γ d -set of T and hence γ d (T ) = S = 2n + S I E (T ) ≥ 2n + γ d (T ) . Clearly, it follows that, γ t (T ) ≥ γ d (T ) . Therefore, by Theorem 1, we get, γ t (T ) = γ d (T ) and hence T ∈ ℑ . Applying the inductive hypothesis on T , we get T ∈ℜ . 83
13. 13. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.5, 2012Case 4: Suppose k ≠ 0 . Then by Lemma 6, k ∈{1,2} and so m = j = 0 . We claim k = 1 . If not, then k = 2 . Wedenote the two components P2 of S ( ex ) by ex and ex in T . Let T = T − ex . Clearly, γ t (T ) = γ t (T ) . Let S be a γd - set of T containing {e w1 } , ew2 ,..., ewn . By Observation 2, {ex , ex } ⊆ S . Since S I E(T ) is a double edge dominating set of T with S I E(T ) = γ d (T) −1, we haveγ t (T ) = γ t (T ) = γ d (T ) > γ d (T ) −1 ≥ γ d (T ) , which is a contradiction to the fact that γ t (T ) ≤ γ d (T ) . Sub case 4.1: For n ≥ 2 . Suppose T i { =T − Un=1 eui ,ei ,ewi } . Then T is obtained from T by type - 3 operation.Now by definition of γ t - set and γd - set, it is easy to observe that γ t (T ) + 2 (n − 1) = γ t (T ) andγ d (T ) + 2(n − 1) = γ d (T ) . Hence γ t (T ) = γ d (T ) and T ∈ ℑ . Applying the inductive hypothesis on T ,T ∈ℜ and hence T ∈ ℜ .Sub case 4.2: For n = 1 . Denote the component P2 of S ( ex ) by ex in T . Suppose S ( e y ) − S [ ex ] has a component H∈{P , P , P } in T , then T = T − S [ ex ] . Therefore we can easily check that T 3 4 6 is obtained fromT by type-6 operation. Now by definition of γd - set, γ d (T ) + 3 = γ d (T ) . For any γ t - set D of T ,D U {e , ew , ex } is a total edge dominating set of T . Clearly, γ t (T ) ≥ γ t (T ) − 3 = γ d (T ) − 3 = γ d (T ) . By Theorem 1, we get γ t (T ) = γ d (T ) and T ∈ ℑ . Applying inductive hypothesis on T , T ∈ ℜ and hence T ∈ℜ . Now if the sub graph S ( ey ) − S [ ex ] has no components P3 , P4 or P6 . Then we consider the structure S ( ey ) in T . By above discussion, S ( ey ) consists of a component P6 = eu e ew ex ex and g number of ofcomponents of P2 , denoted by {e , e ,..., e } . Assume l = 2 . Then, let T 1 2 g = T − S ey  . It can be easily checked  that γ t (T ) + 4 ≥ γ d (T ) = γ d (T ) + 5 , which is a contradiction to the fact that γ t (T ) ≤ γ d (T ) . Henceg ≤ 1. 84
14. 14. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 Suppose T = T − {eu , ex } . Here we can easily check that γ t (T ) + 1 = γ t (T ) . Let S be a γ d - set of T such that S contains as minimum edges of S ey  as possible and S I S [ex ] = {eu , ew ex } . Then   S = ( S − {eu , ew ex }) U {e, ex } is a double edge dominating set of T . Therefore γ t (T ) = γ t (T ) −1 = γ d (T ) −1 = S ≥ γ d (T ) , which implies that γ t (T ) = γ d (T ) where S is a double edge dominating set of T . Hence T ∈ ℑ . Applying inductive hypothesis to T , T ∈ ℜ . If g = 0 , then degT ( ey ) = 2 . Since ex ∉S , to double edge dominate ey , ey ∈S . Therefore ey is in the double edge dominating set D of T . Hence T is obtained from T by type-1 operation. Thus T ∈ℜ . If g = 1 , then degT ( ey ) = 3 . Since ex ∉S to double edge dominate ey , we have ey ∉S and ez ∈S , by the selection of S . Therefore ez is in the double edge dominating set S of T . Hence T is obtained from T by type-1operation. Thus T ∈ ℜ . By above all the Lemmas, finally we are now in a position to give the following main characterization. Theorem 3.10: ℑ = ℜ U { P3} References S. Arumugan and S. Velammal, (1997), “Total edge domination in graphs”, Ph.D Thesis, Manonmaniam Sundarnar University, Tirunelveli, India. F. Harary, (1969), “Graph theory”, Adison-wesley, Reading mass. F. Harary and T. W. Haynes, (2000), “Double domination in graph”, Ars combinatorics, 55, pp. 201-213. T.W. Haynes, S. T. Hedetniemi and P. J. Slater, (1998), “Fundamentals of domination in graph”, Marcel Dekker, New York. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, (1998), “Domination in graph, Advanced topics”, Marcel Dekker, New York. BiographyA. Dr.M.H.Muddebihal born at: Babaleshwar taluk of Bijapur District in 19-02-1959. 85
15. 15. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.5, 2012 Completed Ph.D. education in the field of Graph Theory from Karnataka. Currently working as a Professor in Gulbarga University, Gulbarga-585 106 bearing research experience of 25 years. I had published over 50 research papers in national and international Journals / Conferences and I am authoring two text books of GRAPH THEORY.B. A.R.Sedamkar born at: Gulbarga district of Karnataka state in 16-09-1984. Completed PG education in Karnataka. Currently working as a Lecturer in Government Polytechnic Lingasugur, Raichur Dist. from 04 years. I had published 10 papers in International Journals in the field of Graph Theory. 86