Matching Transversal Edge Domination in Graphs

Transcription

1 Available at htt://vamuedu/aam Al Al Math ISSN: Vol 11, Issue (December 016), Alications and Alied Mathematics: An International Journal (AAM) Matching Transversal Edge Domination in Grahs Anwar Alwardi Deartment of Studies in Mathematics University of Mysore, Manasagangotri Mysuru , India a_wardi@hotmailcom Received: November 19, 015; Acceted: August, 016 Abstract Let G ( V, E) be a grah A subset X of E is called an edge dominating set of G if every edge in E X is adacent to some edge in X An edge dominating set which intersects every maximum matching in G is called matching transversal edge dominating set The minimum cardinality of a matching transversal edge dominating set is called the matching transversal edge domination number of G and is denoted by In this aer, we begin an investigation of this arameter Keywords: Dominating set; Matching set; matching transversal dominating set; Matching transversal domination MSC 010 No: 05C69 1 Introduction By a grah G ( V, E), we mean a finite and undirected grah with no loos and multile edges As usual V and q E denote the number of vertices and edges of a grah G, resectively In general, we use X to denote the subgrah induced by the set of vertices X Nv () and Nv [] denote the oen and closed neighbourhood of a vertex v, resectively for any edge f E The degree of f uv in G is defined by deg(f) = deg(u) + deg(v) The 919

2 90 Anwar Alwardi oen neighbourhood and closed neighbourhood of an edge f in G, denoted by N( f ) and N[ f], resectively, are defined as N( f ) { g V: f and g areadacent} and N[ f ] N( f ) { f } The cardinality of N( f ) is called the degree of f and denoted by deg( f ) The maximum and minimum degree of edge in G are denoted, resectively by and That is, max f E N( f ), min f E N( f ) A set D of vertices in a grah G is a dominating set if every vertex in V D is adacent to some vertex in D The domination number is the minimum cardinality of a dominating set of G A line grah LG ( ) (also called an interchange grah or edge grah) of a simle grah G is obtained by associating a vertex with each edge of the grah and connecting two vertices with an edge, if and only if the corresonding edges of G have a vertex in common The Corona roduct G H of two grahs G and H is obtained by taking one coy of G and VG ( ) coies of H and by oining each vertex of the i -th coy of H to the i -th vertex of G, where 1 i V A matching M in G is a set of airwise non-adacent edges; that is, no two edges share a common vertex The matching number is the maximum cardinality of a matching of G and denoted by 1 A vertex is matched (or saturated) if it is an endoint of one of the edges in the matching; otherwise, the vertex is unmatched A maximal matching is a matching M of a grah G with the roerty that if any edge not in M is added to M, it is no longer a matching; that is, M is maximal if it is not a roer subset of any other matching in grah G In other words, a matching M of a grah G is maximal if every edge in G has a non-ey intersection with at least one edge in M The indeendent transversal domination was introduced by Hamid (01) A dominating set D in a grah G which intersects every maximum indeendent set in G is called indeendent transversal dominating set of G The minimum cardinality of an indeendent transversal dominating set is called indeendent transversal domination number of G and is denoted by it For terminology and notations not secifically defined here we refer reader to Harary (1969) For more details about domination number and its related arameters, we refer to Haynes et al (1998), Samathkumar et al (1985) and Walikar et al (1979) The concet of edge domination was introduced by Mitchell et al (1977) Let G ( V, E) be a grah A subset X of E is called an edge dominating set of G if every edge in E X is adacent to some edge in X In this aer, we introduce the concet of matching transversal domination in a grah and, exact values for some standard grahs, bounds and some interesting results are established Matching Transversal Edge Domination Number Definition 1 Let G = (V, E) be a grah An edge dominating set S which intersects every maximum matching in G is called matching transversal edge dominating set The minimum cardinality of a matching transversal edge dominating set is called the matching transversal edge domination

3 AAM: Intern J, Vol 11, Issue (December 016) 91 number of G and is denoted by γ (G) Theorem 1 For any ath P with, we have,, if or 4; ( P ), if 7; 1, otherwise ( ) {,,, } Obviously ( P) ( P4) Similarly, if 7, easily Let E P v1v vv v 1v we can see that, ( P7 ) Now suose that {,4,7}, then we have three cases Case 1: If 0(b mod), then the set of edges S { v i1v i :0 i 1} is a minimum edge dominating set of P Also the induced subgrah E S K P and every matching in E S contains at most maximum matching of P, then Case : If 1(mod ), then the set of edges edges and 1 ( P ) ( P ) 1 1( P ) 1 i1 i Hence, E S contains no S { v v :1 i } is a minimum edge dominating set of P Furthermore, the induced subgrah E S K P and every matching in 1 E S contains at most edges Now, since 1( P ), it follows 1 that E S contains no maximum matching of P, and then ( P ) ( P ) Case : If (mod ), then the set of edges dominating set of S { v v :0 i } is a minimum edge i1 i P Also, it is easy to check that the induced subgrah and every matching in E S contains at most E S P 1 1 edges and 1 1 1( P ) E S contains no maximum matching of P Thus, ( P ) ( P ) 1 So, Theorem

4 9 Anwar Alwardi For any cycle C of order, we have, =,5; ( C ), otherwise ( ) {,,,, } It is easy to see that ( C) ( C5) Suose that {,5}, then we have three cases: Let E C v1v vv v 1v vv1 Case 1: If 0(mod ), then the set of edges dominating set of in E S contains at most maximum matching of S { v v :0 i } is a minimum edge i1 i C Furthermore, the induced subgrah E S P and every matching edges and 1 1( C ) C, and then ( C) ( C) Case : If 1(mod ), then the set of edges dominating set of 1 i1 i C Also, it is easy to see that the induced subgrah Therefore, every matching in E S contains at most Hence, E S contains no S { v v : 0 i } is a minimum edge 1 edges and E S P 1 1 1( C ) E S contains no maximum matching of C Thus, ( C) ( C) Case : If (mod ), then the set of edges dominating set of E S contains at most matching of Proosition 1 C The induced subgrah 1 edges and C Then ( C) ( C) For any grah G, Observation 1 S { v v :0 i } is a minimum edge i1 i E S K P Hence, Hence, every matching in 1 1( C ) For any grah G with at least one edge, 1 q So, E S contains no maximum A bi-star is a tree obtained from the grah K with two vertices u and v by attaching m endant edges in u and n endant edges in v and denoted by B( m, n )

5 AAM: Intern J, Vol 11, Issue (December 016) 9 The roof of the following roosition is straightforward Proosition For any bi-star grah B mn,, ( Bm, n) min{ m, n} 1 Remark From the definition of and it is easy to observe that ( L), where LG ( ) is the line grah of G Theorem For any Comlete biartite grah K r,s, where r s, ( Kr, s) s it Let G = (V 1, V, E) be any comlete biartite grah K rs,, where V1 r, V s Suose that V 1 = {v 1, v,, v r }, V = {u 1, u,, u s } Let E, where i {1,,, r} is the set of edges which have v i as common vertex It is easy to check that any maximum matching of G contains one edge from each E v i Therefore, any Ev i set of edges is the minimum set which intersects each maximum matching of G and also it is an edge dominating set of G Without loss of generality, let D E { v u, v u, v u,, v u }, where D is an edge dominating set of G v s and intersects every maximum matching of G and with minimum size D s Hence ( K ) s r, s Proosition v i it For any cycle C, ( CP K1) P Theorem 4 If G ( V, E) is a grah with at least one isolated edge, then Let G ( V, E) be a grah with at least one isolated edge say e, and F is a minimum edge dominating set of G Evidently the edge e must belong to any dominating set and also to any matching of G Therefore, the minimum dominating set F intersects every maximum matching of G Hence, F is a minimum matching transversal edge dominating set of G Thus,

6 94 Anwar Alwardi Corollary 1 For any grah = H K, we have γ (G) = γ (G) We can generalize Theorem with the following theorem Theorem (generalized) Let G be a grah with the comonents G1, G,, G r Then, Suose that without loss of generality γ = min 1 i r {γ (G i ) + r =1, i γ (G )} r G1, i G 1 i r G i 1, i G ( ) ( ) min { ( ) ( )} Let S be the minimum matching transversal dominating set of G 1 and let minimum edge dominating set of A be the r G for all It is easy to check that, S ( A ) is a r matching edge dominating set of G Hence, ( G ) ( G ) 1 Conversely, let S be any minimum matching edge dominating set of G Then, S must intersect all the edge set EG ( ) of each comonent G of G and S E( G ) is edge dominating set of G for all 1 Furthermore, for at least one, the set S E( G ) must be matching edge dominating set of G, for otherwise each comonent G will have maximum matching not intersecting the set S E( G ) and hence the union of these maximum matching sets form a maximum matching of G not intersecting S r Hence, S min { ( G ) 1, i ( G )} Theorem 4 1 i r i For any connected grah G with q edges, q, if and only if G K1, 1 or K

7 AAM: Intern J, Vol 11, Issue (December 016) 95 or G K, then it is easy to see that q If G K1, 1 Conversely, let q Then either the minimum edge dominating set of G contains all the edges of G and hence G = mk and in this case since G is connected, then m = 1 which imlies that G K1,1, or the maximum matching has cardinality one, which means G is the comlete biartite grah K1, 1 or K Hence, G K1, 1 or G K Theorem 5 For any connected grah G, γ (G) = 1, if and only if G K If G = K, then it is clear that γ (G) = 1 Conversely, if γ (G) = 1, that means there exists an edge e in G with degree q 1 and e must belong to each maximum matching of G Then G K Theorem 6 Let G be a grah with q edges Then q 1, if and only if G P4 Let q 1 Then the matching number is greater than or equal two If there exist two adacent edges e and f of degree at least two, then S E { e, f } is a matching transversal edge dominating set of G and hence q which is a contradiction Therefore, for any two adacent edges either e or f endant edge Hence G P 4 Conversely, if G = P 4, then it holds that q 1 Theorem 7 Let a and b be any two ositive integers with ba 1 Then there exists a grah G with b edges such that a

8 96 Anwar Alwardi Let b a r, r 1 and let H be any star grah K 1,a with a edges Suose E( H) { e1, e,, e a } Let G be the grah obtained from H by attaching r 1 edges at the endant vertex of the edge e 1, and one endant edge at each endant vertex of the edge e i for i Let f i, where i be the endant edges in G adacent to ei, i It is easy to check that a and the set D { e1, f,, f a } is the minimum edge dominating set and D intersectd all the maximum matching of G Hence, a and E b Theorem 8 Let G ( V, E) be a grah with a unique maximum matching Then 1 Let G be a grah with a unique maximum matching S and let D be a minimum edge dominating set of G Let e be any edge in S Then D {} e is a matching transversal edge dominating set Hence 1 The converse of Theorem 8, is not true in general, for examle for G C4 we have that, but there are two maximum matching sets Theorem 9 For any grah G, we have Let f be an edge with degree deg( f) and let S be a minimum edge dominating set of G Every maximum matching of G contains an edge of N[ f ] Hence, S N[ f ] is a matching transversal edge dominating set of G We have also S intersects N[ f ], and it follows that S N[ f ] Hence, Let F 1 be the 5-vertex ath, F the grah obtained by identifying a vertex of a triangle with one end vertex of the -vertex ath, F the grah obtained by identifying a vertex of a triangle with a vertex of another triangle (See Figure 1)

9 AAM: Intern J, Vol 11, Issue (December 016) 97 F Figure 1: The family F1, F and F Theorem 10 (Raman et al (009)) If diam and if none of the three grahs F1, F and F of Figure l, is an induced subgrah of G, then diam( L) Theorem 11 Let G be a grah with diameter diam and none of the three grahs F1, F and F in Figure 1 is an induced subgrah of G Then 1 Let f be an edge with deg( f) Since diam it follows from Theorem 10 that diam( L) Then N[ f ] is an edge dominating set of G and every maximum matching contains an edge of N[ f ] Therefore, N[ f ] is a matching transversal edge dominating set Hence, 1 Theorem 1 For any ( q, ) grah G, q, if and only if each comonent of G is isomorhic to K Let q Then by (Jayaram, SR (1987)), G has q comonents, each

10 98 Anwar Alwardi of which is isomorhic to a star Also, since q, then each comonent contains only one edge Hence, each comonent of G is isomorhic to K Conversely, if each comonent of G is isomorhic to K, it is easy to check that q Conclusion In this aer, we define a new domination invariant called matching transversal edge domination in grahs which mixes the two imortant concets in Grah Theory, domination and matching Some imortant roerties and exact values for some standard grahs are obtained Also some bounds for this arameter and some relations with the other domination arameters are established As this aer is the first to investigate this arameter still there are many things to study in the future, like how to artition the edges into matching transversal edge domination sets with maximum size, how this arameter changes and unchanges by deleting edges, and similar to the standard domination, we can study this new arameter by studying total, connected, indeendent, slit, nonslit matching transversal edge domination in grahs Acknowledgment: The author is thankful to anonymous referees and the Editor-in-Chief, Professor Aliakbar Montazer Haghighi for useful comments and suggestions towards the imrovements of this aer REFERENCES Bondy, J and Murthy U (1976) Grah Theory with alications, North Holland, New York Godsil, C and Royle G (001) Algebraic grah theory, Vol 07 of Graduate Texts in Mathematics, Sringer-Verlag, New York Hamid, IS (01) Indeendent transversal Domination in Grahs, Grahs,Discussions Mathematicae Grah Theory, 5-17 Harary, F (1969) Grah theory, Reading Mass Haynes, TW, Hedetniemi ST and Slater, PJ (1998) Fundamentals of domination in grahs, Marcel Dekker, Inc, New York Jayaram, SR (1987) Line domination in grahs, Combin, 57-6 Mitchell, S, Hedetniemi, ST (1977) Edge domination in trees, Grah Theory and comuting, Congr Number, 19, Raman, HS, Revankar, DS, Gutman, I, and Walikar, HB (009) Distance sectra and distance energies of iterated line grahs of regular grahs, Publ Inst Math (Beograd),85, 9 46 Samathkumar, E, and Neeralagi, PS (1985) The neighborhood number of a grah, Indian J

11 AAM: Intern J, Vol 11, Issue (December 016) 99 Pure and Al Math, 16 (), 16-1 Walikar, HB, Acharya, BD, and Samathkumar, E (1979) Recent develoments in the theory of domination in grahs, Mehta Research Institute, Allahabad, MRI Lecture Notes in Math,1