The Dominating Grah (G) of a Grah G M. Bhanumathi 1, J. John Flavia 2 1 Associate Professor, Government Arts College for Women, Pudukkottai, TN, India 2 Research Scholar, Government Arts College for Women, Pudukkottai, TN, India 1 bhanu_ks@yahoo.com, 2 flavimaths@gmail.com Abstract: The dominating grah (G) of a grah G is obtained from G with vertex set V = V(G) S, where V = V(G) and S is the set of all minimal dominating sets of G. Two elements in V are said to satisfy roerty a if u, v V and are adacent in G. Two elements in V are said to satisfy roerty b if u = D 1, v = D 2 S and have a common vertex. Two elements in V are said to satisfy roerty c if u V(G), v = D S such that u D. Two elements in V are said to satisfy roerty d if u, v V(G) and there exists D S such that u, v D. A grah having vertex set V and any two elements in V are adacent if they satisfy any one of the roerty b, c, d is denoted by (G). In this aer, we have studied some basic roerties of (G). We have characterized grahs G for which (G) has some secific roerties. Also, we have established some extremal roerties of (G). Keywords: Dominating set, minimal dominating set, dominating grah. 2010Mathematical Classification Code: 05C12, 05C69. 6 I. INTRODUCTION Grahs discussed in this aer are undirected and simle. For grah theoretic terminology refer to Harary[3], Buckley and Harary[1]. For a grah, let V(G) and E(G) denotes its vertex and edge set resectively. A grah with vertices and q edges is called a (, q) grah. The degree of a vertex v in a grah G is the number of edges of G incident with v and it is denoted by deg(v). The length of any shortest ath between any two vertices u and v of a connected grah G is called the distance between u and v and it is denoted by d G (u, v). The distance between two vertices in different comonents of a disconnected grah is defined to be. For a connected grah G, The eccentricity e(v) of v is the distance to a vertex farthest from v. Thus, e(v) = max{d(u, v) : u V}.The radius rad(g) = r(g) is the minimum eccentricity of the vertices, whereas the diameter diam(g) is the maximum eccentricity. If these two are equal in a grah, that grah is called self-centered grah with radius r and is called an r self-centered grah. For any connected grah G, rad(g) diam(g) 2rad(G). v is a central vertex if e(v) = r(g). The center C(G) is the set of all central vertices. For a vertex v, each vertex at a distance e(v) from v is an eccentric vertex of v. The girth g(g) of the grah G, is the length of the shortest cycle(if any) in G. A grah G is connected if every two of its vertices are connected, otherwise G is disconnected. The vertex connectivity or simly connectivity κ(g) of a grah G is the minimum number of vertices whose removal from G results in a disconnected or trivial grah. The edge connectivity λ(g) of a grah G is the minimum number of edges whose removal from G results in a disconnected or trivial grah. A set S of vertices of G is indeendent if no two vertices in S are adacent. The indeendence number β o (G) of G is the maximum cardinality of an indeendent set. The concet of distance in grah lays a dominant role in the study of structural roerties of grahs in various angles using related concet of eccentricity of vertices in grahs. The concet of domination in grahs was introduced by Ore[8]. The concets of domination in grahs originated from the chess games theory and that aved the way to the develoment of the study of various domination arameters and its relation to various other grah arameters. For details on γ(g), refer to [2, 9]. A set D V is said to be a dominating set in G, if every vertex in V D is adacent to some vertex in D. The cardinality of minimum dominating set is called the domination number and is denoted by (G). A dominating set D is called a minimal dominating set if no roer subset of D is a dominating set. The uer domination number Г(G) of a grah G is the maximum cardinality of a minimal dominating set of G. Domatic number d(g) of a grah is the largest order of a artition of V(G) into dominating set of G. For a given grah G, the end edge grah G + is the grah obtained from G by adoining a new edge u i u i at each vertex u i of G in such a way that this edge has exactly one vertex u i in common with G. In [5, 6, 7], Kulli, Janakiram and Niranan introduced the following concets in the field of domination theory. The minimal dominating grah MD(G) of a grah G is the intersection grah defined on the family of all minimal dominating sets of vertices of G[5]. The common minimal dominating grah CD(G) of a grah G is the grah having the same vertex set as G with two vertices adacent in
CD(G) if and only if there exists a minimal dominating set in G containing them [6]. The dominating grah D(G) of a grah G = (V, E) is a grah with V(D(G)) = V S, where S is the set of all minimal dominating sets of G and with two vertices u, v V(D(G)) adacent if u V and v = D is a minimal dominating set of G containing u[7]. In this aer, we define a new dominating grah (G) with roerty b, c and d. we find some basic roerties of (G). Also, we characterize grahs G for which (G) has some secific roerties. We need the following results to study the dominating grah (G) of a grah G. Theorem: 1.1[5] For any grah G, MD(G) is comlete if and only if G contains an isolated vertex. Theorem: 1.2[4] A grah G is Eulerian if and only if every vertex of G is of even degree. Theorem: 1.3[4] If for all vertices v of G, deg(v) /2 where 3, then G is Hamiltonian. II. THE DOMINATING GRAPH BCD (G) OF A GRAPH G We define a new class of intersection grahs in the field of domination theory as follows. Definition: 2.1 A grah having vertex set V = V(G) S, where S is the set of all minimal dominating sets of G. Two elements in V are said to satisfy roerty b if u = D 1, v = D 2 S and have a common vertex. Two elements in V are said to satisfy roerty c if u V(G), v = D S such that u D. Two elements in V are said to satisfy roerty d if u, v V(G) and there exists D S such that u, v D. A grah having vertex set V and any two elements in V are adacent if they satisfy any one of the roerty b, c, d is denoted by (G). Here, the elements of V(G) are called as oint vertices and the elements of S are known as set vertices. Clearly, MD(G) (G), CD(G) (G), D(G) (G). Examle: G: {1, 3}, {1, 4}, {2, 3}, {2, 4} are minimal dominating sets of G. G (G): 7 1 2 3 4 Remark 2.1: If there is a minimal dominating set having m vertices then (G) has K m+1 as an induced sub grah. So K Г(G)+1 is always an induced sub grah of (G), where Г(G) is the cardinality of the minimal dominating set with maximum number of vertices. Remark 2.2: For any grah G, +d(g) ( 1) 2, where d(g) is the domatic number of G and denotes the number of vertices of (G). (i) For any grah G, 2 q ( 1)+ S ( s 1)/2, where q denoted the number of edges in (G). (ii) deg (G) v = 1 1 D, where D denote the minimal dominating sets containing v in G. (iii) deg D i S 1+ D i, 1 i n. Theorem 2.1: If G = K, then (G) is K 2. Proof: When G = K. Each vertex forms a minimal dominating set. Hence we get such minimal dominating sets. By the definition of (G), (G) is K 2. Theorem 2.2: If G = K 1, 1, then (G) is K 2 K. Proof: Let G = K 1, 1. Let v be the vertex of degree 1. Since K 1, 1 has exactly two minimal dominating sets. {v} and all the remaining endent vertices form minimal dominating sets. By the definition of (G), {v} is adacent to exactly one corresonding vertex of G, and all the remaining vertices are adacent to another minimal dominating set and each vertex is adacent to other vertices. Thus, the resulting grah will be K 2 K. Therefore, (G) = K 2 K. Corollary 2.1: β o ( (G)) = 2. We characterize the grahs for which (G) is comlete.
Theorem 2.3: (G) is a comlete grah with +1 vertices if and only if G = K. Proof: Suose G = K. The whole vertex set form a minimal dominating set of G. In (G), all vertices of G are adacent to each other and also adacent to the minimal dominating set. Thus, we get a comlete grah with +1 vertices. Therefore, (G) = K +1. Conversely, suose (G) is comlete. Then each vertex of (G) is adacent. This is ossible only the whole vertex set form minimal dominating set. Thus, G = K. Theorem 2.4: For any grah G, G (G). Proof: If u and v are not adacent in G. Then {u, v} form an indeendent set. Thus there exists a minimal dominating set containing both u and v. Thus, it follows that G (G). We obtain a necessary and sufficient condition on a grah G such that (G) is connected. Theorem 2.5: For any grah G with vertices, 2, (G) is connected if and only if (G) < 1. Proof: Let (G) < 1. To rove that (G) is connected. Let u, v be any two vertices of G. If u and v are not adacent in G, then by theorem 2.4, u and v are adacent in (G). If u and v are adacent in G and some vertex w V(G) distinct from both u and v and is adacent to neither u nor v. Then there exists two minimal dominating sets D 1 and D 2 such that D 1 and D 2 contain u, w and v, w resectively. Thus u and v are connected in (G) by the aths u-w -v, v-d 2 -w-u and u-d 1 -w-v and D 1, D 2 are adacent. Conversely, suose, (G) is connected. To rove that (G) < 1. Suose (G) = 1 and u is a vertex of degree 1. Then {u} = D is a minimal dominating set of G. Since G has at least two vertices, u is adacent to all other vertices of G. Thus V D contains minimal dominating sets. In (G), there is no ath oining u to any vertex of V D. This imlies that (G) is disconnected, a contradiction. Hence, (G) < 1. Corollary 2.2: If (G) = 1, K 2 is a comonent of (G). We characterize the grahs for which (G) is a tree. Theorem 2.6: (G) is a tree if and only if G = K 1. Proof: Suose G = K 1, then by the definition of (G), (G) is K 2. Conversely, suose (G) is a tree. To rove that G = K 1. Assume G K 1. By theorem 2.5, (G) is connected if and only if (G) < 1. Let (G) < 1. Then there exists a minimal dominating set D contains u and v, from the definition of (G), D, u and v form a triangle, a contradiction. Thus, G = K 1. Next radius and diameter of (G) is found out and classification of grahs such as (G) is self-centered with diameter two is discussed. Theorem 2.7: Let G be a connected grah with (G) < 1, then in (G), d(u, v) 3. Proof: Since V = V(G) S, where S is the set of all minimal dominating sets of G. Let u, v V. Consider the following cases. Suose u, v V(G), d G (u, v) > 2. Then there exists a minimal dominating set D 1 such that D 1 contains u and v. Thus, in (G), d(u, v) = 1. Suose d G (u, v) > 2, u D 2 and v D 3. Then there exists a vertex z V(G) is adacent to both u and v such that z D 2 and D 3. Thus, it follows that, in (G), d(u, v) = d(u, z)+d(z,v) = 2. Suose d G (u, v) = 1, u D 4 and v D 5. Then there exists a vertex w V(G) is neither adacent to u nor v such that w D 4 and D 5. This imlies that, in (G), d(u, v) = d(u, w)+d(w, v) = 2. Suose u V(G) and v V(G). Then v = D is a minimal dominating set of G. If u D, then u and v are adacent in (G). Suose u D. Consider the following two sub cases. Sub If u D, then there exists a vertex w D such that w D. Thus in (G), d(u, v) = d(u, w)+d(w, v) = 2 (or) d(u, D ) = d(u, D )+d(d, D ) = 2. Sub Suose u lies on only one minimal dominating set D. If there exists a minimal dominating set D 6 such that D 6 is adacent to both D and D. Thus, it follows that, in, d(u, D ) = d(u, D )+d(d, D 6 )+d(d 6, D ) = 3. Case (iii) 8
Suose u, v V(G). Then u = D 7 and v = D 8 are two minimal dominating sets of G. If D 7 and D 8 are not disoint, then in (G), d(u, v) = d(d 7, D 8 ) = 1. Otherwise, there is a minimal dominating set D 9 such that D 9 is adacent to both D 7 and D 8. Thus, it follows that, in (G), d(u, v) = d(d 7, D 8 ) = d(d 7, D 9 )+(D 9, D 8 ) = 2. From the above cases, d ( u, v ) 3 is established. ( G) Remark 2.2: If rad(g) > 1, By theorems 2.5 and 2.7, (G) is connected and diam( (G)) 3. Theorem 2.8: Let G be a grah with an isolated vertex. Then rad( (G)) = 1 and diam( (G)) = 2. Proof: Suose G has an isolated vertex. Let u be an isolated vertex and u lies on all the minimal dominating sets of G. Suose v( u) V(G), we can form a minimal dominating set D such that v D. Thus all vertices in (G) are adacent to u. Therefore eccentricity of u in (G) is one. Let w, v V(G). Suose v D 1 and w D 2. Since, in this case every minimal dominating set is adacent to each other. Thus, in (G), d(v, w) = d(v, u)+d(u, w) = 2 and d(d i, D ) = 1 for i. Suose x V(G) and y V(G). Then y = D 3 is a minimal dominating set of G. If x D 3. Then in (G), d(x, y) = d(x, u)+d(u, y) = 2. Therefore, eccentricity of oint vertices is 2 excet u and eccentricity of set vertices is 2. Hence, rad( (G)) = 1 and diam( (G)) = 2. Theorem 2.9: Let G be a disconnected grah without isolated vertices. Then (G) is 2- self-centered. Proof: Suose G is a disconnected grah. Then G has at least two comonents G 1 and G 2. Consider the following cases Let u V(G 1 ), v V(G 2 ). Then there exists a minimal dominating set D such that D contains u and v. Thus, in (G), d(u, v) = 1. Suose x, y V(G 1 ) and x D 1, y D 2. Then there exists a vertex z V(G 2 ) such that z D 1 and z D 2 and vice versa. Thus, it follows that, in (G), d(x, y) = d(x, z)+d(z, y) = 2. Suose a V(G 1 ), b S. Then b = D 3 is a minimal dominating set of G. If a D 3, then there exists a vertex c from any other comonents such that c D 3 and c is adacent to a, it follows that, in (G), d(a, b) = d(a, c)+d(c, b) = 2. Case (iii) Suose u, v S. Then u = D 4 and v = D 5. If D 4 and D 5 are not disoint, then in (G), d(d 4, D 5 ) = 1. Suose D 4 and D 5 are disoint. Then there exists a minimal dominating set D such that D is adacent to both D 4 and D 5. Thus, it follows that, in (G), d(d 4, D 5 ) = d(d 4, D )+d(d, D 5 ) = 2. Hence, eccentricity of oint vertices is 2 and eccentricity of set vertices is also 2. Therefore, (G) is 2 selfcentered. Theorem 2.10: If D 1, D 2 V(G) such that D 1 and D 2 are minimal dominating sets and there exists v D 1 (or D 2 ) such that there is a minimal dominating set containing v and elements of D 2 (or D 1 ), then (G) is 2- selfcentered. Proof: Let G be a connected grah with rad(g) > 1. Suose D 1 and D 2 are two minimal disoint dominating sets of G, and there exists v D 1 (or D 2 ) such that there is a minimal dominating set D 3 containing v and elements of D 2 (or D 1 ). Thus, in (G), d(v, D 1 (ord 2 )) = d(v, D 3 )+d(d 3, D 1 (ord 2 )) = 2. Let v be the common element of D 1 (ord 2 ) and D 3. Since v and v are adacent in (G). Thus, it follows that, d(v, D 1 (ord 2 )) = d(v, v )+d(v, D 1 (or D 2 )) = 2. By case (i) and case (iii) of theorem 2.7, eccentricity of all oint vertices and set vertices are 2. Hence, (G) is 2- self-centered. Theorem 2.11: If D 1, D 2 V(G) such that D 1 and D 2 are minimal disoint dominating sets and there exists v D 1 (or D 2 ) such that there is no minimal dominating set containing v and elements of D 2 (or D 1 ), then diam( (G)) = 3. Proof: Let G be a connected grah with rad(g) > 1. Suose D 1 and D 2 are two minimal disoint dominating sets and there exists v D 1 (or D 2 ) such that there is no minimal dominating set containing v and elements of D 2 (or D 1 ). Consider the following cases. There exists a vertex z V(G) such that z D 1 (or D 2 ) and z D 3 which is adacent to D 1 (or D 2 ). Thus, it follows that, in (G), d(v,d 1 (or D 2 )) = d(v, z)+d(z, D 3 )+d(d 3, D 1 (or D 2 )) = 3. There exists vertices x, y V(G) such that x D 1 (or D 2 ), y D 1 (or D 2 ) and {x, y} D 4 is the minimal dominating set of G. Thus, it follows that, in (G), d(v, D 1 (or D 2 )) = d(v, x)+d(x, y)+d(y, D 1 (or D 2 )) = 3. 9
There exists a minimal dominating set D 5 such that D 5 is adacent to both D 1 and D 2. Thus, it follows that, in (G), d(v, D 1 (or D 2 )) = d(v, D 1 (or D 2 ))+d(d 1 (or D 2 ), D 3 )+d(d 3, D 1 (or D 2 )) = 3. Suose u, u V(G). Then there exists a minimal dominating set D such that D contains u, u. Then, it follows that, in (G), d(u, u ) = 1. Suose u D and v D. Then there exists a vertex x V(G) such that x D and D. Thus, if follows that, in (G), d(u, u ) = d(u, x )+d(x, u ) = 2. Case (iii) Suose u, u V(G). Then u = D 4 and u = D 5 are two minimal dominating sets of G. If D 4 and D 5 have a common vertex, then in (G), d(d 4, D 5 ) = 1. Suose D 4 and D 5 are disoint. Then there exists a minimal dominating set D such that D is adacent to both D 4 and D 5. Thus, it follows that, in (G), d(d 4, D 5 ) = d(d 4, D )+d(d, D 5 ) = 2. So, in all the cases, diam( (G)) is 3. Corollary 2.3: If G with rad(g) > 1 is a connected grah and there exists a vertex v V(G) such that v is in exactly only one minimal dominating set, then diam( (G)) = 3. Proof: Proof follows from theorem 2.11. Theorem 2.12: (G) = K +1 if and only if G = 10 K. Proof: Suose G = K, then by theorem 2.3 (G) is a comlete grah with +1 vertices. Conversely, suose (G) = K +1. Then rad( (G)) = diam( (G)) = 1. To rove that G = K. On the contrary, assume that G K, G is connected. Then there exists at least two minimal dominating sets and ( (G)) < 1, where is the number of vertices in (G), a contradiction. Hence, G must be K. Theorem 2.13: rad( (G)) = 1, diam( (G)) = 2 if and only if G has at least one isolated vertex. Proof: Suose G has an isolated vertex. Then by theorem 2.8 rad( (G)) = 1, diam( (G)) = 2. Conversely, suose rad( (G)) = 1, diam( (G)) = 2. To rove that G has at least one isolated vertex. On the contrary, assume that G is disconnected without isolated vertices. Then, by theorem 2.9, (G) is 2- self-centered. Hence G has at least one isolated vertex. Theorem 2.14: (G) is 2- self-centered if and only if G is any one of the following: (i) G is a disconnected grah without isolated vertices. (ii) G is a connected grah. For D 1, D 2 V(G) such that D 1 and D 2 are two minimal disoint dominating sets of G and there exists v D 1 (or D 2 ) such that there exists a minimal dominating set containing v and elements of D 2 (or D 1 ). Proof: Proof follows from theorems 2.9 and 2.10. Theorem 2.15: rad( (G)) = 2, diam( bc (G)) = 3 if and only if D 1, D 2 V(G) such that D 1 and D 2 are two minimal disoint dominating sets of G and there exists v D 1 (or D 2 ) such that there is no minimal dominating set containing v and elements of D 2 (or D 1 ). Proof: Proof follows from theorem 2.11. Following two theorems deals with the domination arameters of (G). Theorem 2.16: γ( (G)) = if and only if G = K. Proof: Suose G = K. By the definition of (G), (G) is K 2. Thus, if follows that γ( (G)) =. Conversely, suose γ( (G)) =. To rove that G = K. On the contrary, assume that G K. Then there exists at least two non-adacent vertices u and v in G. There exists a minimal dominating set D such that D contains u and v. The remaining vertices other than u and v with D form minimal dominating set of (G). This imlies that γ( (G)) 1, a contradiction. Hence G = K. Theorem 2.17: For any grah G, 1 γ( (G)). The both uer and lower bounds are shar. Proof: Proof follows from theorems 2.16 and 2.17. Theorem 2.18: If G = K K 1, for all 2, then K + and K 1, are edge disoint sub grahs of (G). Proof: Let G = K K 1. Let v i be an isolated vertex in G. Then deg(v i ) is 2 in (G). Since MD(G) is an induced sub grah of (G). From theorem 1.1, it follows that K is a sub grah of (G). Clearly, K + and K 1, 2 are edge disoint sub grahs of (G). Next we find out the girth of (G). Theorem 2.19: Let G be a connected grah with (G) < 1, then girth of (G) is three. Proof: Let G be a connected grah with (G) < 1. Then (G) is connected. If there is a minimal dominating
set having m vertices then (G) has K m+1 as an induced sub grah. Therefore girth of (G) is three. Next we study the connectivity and edge connectivity of (G). Theorem 2.20: For any grah G, κ( abc (G)) min{min{ deg ( v ) ( G ) i, 1 i, min{ deg ( G) S minimal dominating sets of G., 1 n}}, where S s are the Proof: Let G be a (, q) grah. We consider the following cases. Let v V(G) having minimum degree among all v i s. If the degree of v is less than degree of any other vertex in (G). Then by deleting those vertices of (G), which are adacent to v, the resulting grah is connected. Let u = S, having minimum degree among all vertices of S s. If the degree of u is less than degree of any other vertex in (G). Then by deleting those vertices of (G), which are adacent to u, the resulting grah is disconnected. Hence, κ( abc (G)) min{min{ deg ( v ) ( G ) i, 1 i, min{ deg ( G) S, 1 n}}. Theorem 2.21: For any grah G, λ( abc (G)) min{min{ deg ( v ) ( G ) i, 1 i, min{ deg ( G) S dominating sets of G., 1 n}}, where S s are the minimal Proof: Proof is similar to theorem 2.21. Next two theorems give the traversability roerties of the grah (G). Theorem 2.22: Let (G) < 1 and Г(G) = 2. If every vertex is in exactly two minimal dominating sets, then (G) is Hamiltonian and also Eulerian. Proof: Let (G) < 1 and Г(G) = 2. Then (G) is connected and γ(g) = Г(G). Every minimal dominating set contains two vertices, suose every vertex is in exactly two minimal dominating sets. Then v V(G) and D is a minimal dominating set of G, deg v deg D = 4. Then by theorem ( G) ( G) 1.2, (G) is Eulerian. Since for each v V(G), there exists two minimal dominating set containing v. Then (G) contains a sanning cycle. Hence it is Hamiltonian. Theorem 2.23: If G is a ( 2) regular grah, then (G) is Eulerian and also Hamiltonian. Proof: Let G be a ( 2) regular grah. Then any two vertices form the minimal dominating set of G. In (G), degree of oint vertices is 2( 1) and degree of set vertices is also 2( 1). Thus, every vertex of (G) has an even degree. Then, it follows that (G) is Eulerian. In (G), v V, deg v /2. Thus, it follows that (G) is Hamiltonian. III. CONCLUSION In this aer, we have defined and studied the new dominating grah (G). Eccentricity of vertices of (G) is found out. Classification of grahs for which (G) is self-centered with diameter two is discussed. Proerties such as traversability, connectivity and edge connectivity of (G) are established. Domination arameters of (G) is also studied. IV. REFERENCES [1] Buckley. F, Harary. F, Distance in grahs, Addison Wesley, Publishing Comany (1990). [2] Cockayne, E.J., Hedetniemi, S.T., Towards a theory of domination in grahs, Networks: 47-261,1977. [3] Harary, F., Grah theory, Addition - Wesley Publishing Comany Reading, Mass (1972). [4] Kulli V.R, Theory of domination in grahs,vishwa International ublications (2010). [5] Kulli V.R, Janakiram B, The minimal dominating grah, Grah theory notes of New York, XXVIII, 12-15, 1995. [6] Kulli V.R, Janakiram B, The common minimal dominating grah, Indian J. Pure al. Math., 27(2): 193-196, Februrary 1996. [7] Kulli V.R, Janakiram B, Niranan K, The dominating grah, New York Academy of Sciences, Grah Theory Notes of New York XLVI, 5-8 (2004). [8] O. Ore, Theory of grahs, Amer. Math. Soc. Colloq. Publ., 38, Providence(1962). [9] Teresa W. Haynes, Stehen T. Hedetniemi, Peter J. Slater, Fundamentals of domination in grahs, Marcel Dekker, New York, 1998. 11