Edge Fixed Steiner Number of a Graph

International Journal of Mathematical Analysis Vol. 11, 2017, no. 16, 771-785 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.7694 Edge Fixed Steiner Number of a Graph M. Perumalsamy 1, P. Arul Paul Sudhahar 2, J. John 3 and R. Vasanthi 1 1 Department of Mathematics Alagappa Chettiar College of Engineering and Technology Karaikudi-3, Tamilnadu, India 2 Department of Mathematics Rani Anna Government College for Women Tirunelveli-8, Tamilnadu, India 3 Department of Mathematics Government College of Engineering, Tirunelveli Tamilnadu, India Copyright c 2017 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For a non-empty set W of vertices in a connected graph G, the Steiner distance d(w) of W is the minimum size of a connected subgraph of G containing W. Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to W or a Steiner W-tree. S(W) denotes the set of vertices that lies in Steiner W-trees. Let G be a connected graph with at least 2 vertices. A set W V(G) is called a Steiner set of G if S(W) = V(G). The Steiner number s(g) is the minimum cardinality of a Steiner set. Let G be a connected graph with at least 3 vertices. For an edge e = xy in G, a set W V(G) {x, y} is called an edge fixed Steiner set of G if W = W {x, y} is a Steiner set of G. The minimum cardinality of an edge fixed Steiner set is called the edge fixed Steiner number of G and is denoted by s e (G). Also the Steiner W-tree necessarily contains the edge e and is called edge fixed Steiner W-tree. In this paper, we begin with an investigation of this parameter.

772 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi Keywords: Steiner set, edge fixed Steiner set, Steiner number, edge fixed Steiner number, diameter of a graph 1 Introduction By a graph G = (V, E), we mean a finite undirected connected graph without loops or multiple edges. The order and size of G are denoted by p and q respectively. The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. An u-v path of length d(u,v) is called an u-v geodesic. It is known that the distance is a metric on the vertex set of G. For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. The minimum eccentricity among the vertices of G is called the radius and the maximum eccentricity is called the diameter of G and are denoted by rad G and diam G respectively. For basic graph theoretic terminology, we refer to Harary[5]. For a non-empty set W of vertices in a connected graph G, the Steiner distance d(w) of W is the minimum size of a connected subgraph of G containing W. Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to W or a Steiner W-tree. A set W V of vertices in the graph G is called a Steiner set if every vertex in G lies in a Steiner-W-tree which is a minimum connected subgraph of G containing W. The Steiner number s(g) is the minimum cardinality of a Steiner set. A Steiner set with minimum cardinality is denoted as s-set. The edge fixed geodomination number of a graph was introduced by Santhakumaran and Titus[10]. In this paper, we introduce edge fixed Steiner number of a graph using Steiner sets in a graph. We provide the edge fixed Steiner number of some standard graphs. Also,we provide the bounds for edge fixed Steiner number through several parameters. For positive integers r, d and n 2 with r d 2r, there exists a connected graph G with rad G = r, diam G = d and s e (G) = n or n 1 for any edge e = xy in G. Also for any two positive integers a and b with 2 a b, there exists a connected graph G with s(g) = a and s e (G) = b for some edge e = xy in G. The follwoing theorems are used wherever required. Theorem 1.1. [3] Each extreme vertex of a graph G belongs to every Steiner set of G. In particular,each end vertex of G belongs to every Steiner set of G. Theorem 1.2. [3] Every non trivial tree with exactly k end vertices has Steiner number k.

Edge fixed Steiner number of a graph 773 Theorem 1.3. [3] For any cycle C P (p 3), s(c P ) = { 2 if p is odd 3 if p is even 2 Edge fixed Steiner number Definition 2.1. Let G be a simple connected graph with at least three vertices. For an edge e = xy in G, a set W V(G) {x, y} is called an edge fixed Steiner set of G if W = W {x, y} is a Steiner set of G. The minimum cardinality of an edge fixed Steiner set is called the edge fixed Steiner number of G and is denoted by s e (G). An edge fixed Steiner set with minimum cardinality is denoted by s e -set. Example 2.2. Consider the graph G shown in Figure 1. v 2 v 3 v 1 v 4 v 6 v 5 Figure 1: The minimum edge fixed Steiner sets for each edge e of G and the corresponding edge fixed Steiner numbers of G are given in the following table. Edge e s e -sets s e (G) v 1 v 2 {v 4, v 5 } 2 v 2 v 3 {v 1, v 4, v 5, v 6 } 4 v 3 v 4 {v 1, v 6 } 2 v 4 v 5 {v 1, v 2 } 2 v 5 v 6 {v 1, v 2, v 3, v 4 } 4 v 6 v 1 {v 3, v 4 } 2 v 2 v 6 {v 1, v 4 } 2 v 3 v 5 {v 1, v 4 } 2 Theorem 2.3. If G is any connected graph G of order p and e is an edge of G, then 1 s e (G) p 2. Proof: Let G be a connected graph with at least three vertices. Let e = xy be an edge of G and let W be an edge fixed Steiner set of G.

774 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi Then W V(G) {x, y}. Therefore s e (G) 1. Also since the vertices x and y do not belong to any edge fixed Steiner set W, it follows that s e (G) p 2. Therefore 1 s e (G) p 2 for every edge e in G. Remark 2.4. The bounds for s e (G) are sharp. { 1 if e is an end edge of Pp Theorem 2.5. For the path P p with p 3, s e (P p ) = 2 otherwise Proof: Let V (P p ) = {v 1, v 2,..., v p }. Case 1: Let e be an end edge of P p. Then e = v 1 v 2 or v p 1 v p. If e = v 1 v 2, then {v p } is the edge fixed Steiner set of P p. Similarly, if e = v p 1 v p, then {v 1 } is the edge fixed Steiner set of P p. Therefore s e (P p ) = 1. Case 2: Let e be any internal edge of P p. Then {v 1, v p } is the edge fixed Steiner set of P p. Therefore s e (P p ) = 2. Theorem { 2.6. If C p is a cycle of order p and e is any edge of C p, then 1 if p is odd s e (C p ) = 2 if p is even Proof: Let e = xy be any edge of C p. Case 1: Let p be odd. Choose a vertex u such that d(x, u) = d(y, u) = p 1. Then {u} is an edge 2 fixed Steiner set of C p. Therefore s e (C p ) = 1. Case 2: Let p be even. It is easily verified that no singleton set of C p is an s e -set of C p and so s e (C p ) 2. Let u and v be the antipodal vertices of x and y respectively. Then W = {x, y} is an edge fixed Steiner set of G so that s e (C p ) = 2. Theorem 2.7. For any complete graph K p with p 3, s e (K p ) = p 2 where e is any edge of K p. Theorem 2.8. Let e = xy be any edge of a connected graph G of order at least 3. Then every extreme vertex of G, other than the vertices x and y (whether x and y are extreme vertices or not) belongs to every edge fixed Steiner set in G. In particular, every end vertex of G other than x and y belongs to every edge fixed Steiner set of G.

Edge fixed Steiner number of a graph 775 Proof: Let e = xy be any edge of G and let W be an edge fixed Steiner set of G. By the definition of edge fixed Steiner set, x and y do not belong to it. So let v x, v y be an extreme vertex of G. Let W =W {x, y}. Then W is a Steiner set of G. Suppose that v / W. Then v / W. Since W is a Steiner set of G, the vertex v lies on a Steiner-W -tree T so that W V(T) and v V(T). Let deg T v = k. If k = 1, then v is an end vertex of the Steiner-W -tree T. Since every end vertex of a Steiner-W -tree belongs to W, it follows that v W which is a contradiction. Hence v W. If k 2, let N T (v) = {u 1, u 2,...,u k } be the neighbourhood of v in T. Since v is an extreme vertex of G, it follows that u i u j E(G) for all i, j with 1 i, j k and i j. Let T be the tree in G obtained from T by deleting the vertex v and adding k 1 edges u i u i+1, 1 i k 1. Then W V(T ) and V (T ) = V (T ) 1. This is a contradiction since T is a Steiner-W -tree. Hence v W. Corollary 2.9. If G is a connected graph with k extreme vertices and e = xy is any edge of G, then (i) max.{1, k} s e (G) p 2 if neither x nor y is an extreme vertex of G (ii) max.{1, k 1} s e (G) p 2 if either x or y is an extreme vertex of G (iii) max.{1, k 2} s e (G) p 2 if both x and y are extreme vertices of G. Corollary 2.10. Let T be any non trivial tree and k be the number of end vertices in T. Let e = xy be any edge of T. Then { k if neither x nor y is an end vertex of T s e (T ) = k 1 if either x or y is an end vertex of T Proof: Case 1: Let e = xy where neither x nor y is an end vertex of T. Let W = {u 1, u 2,...,u k } be the set of all end vertices of T. Then by Corollary 2.9(i), s e (T ) k. Also it is clear that W is an edge fixed Steiner set of T so that s e (T ) k. Therefore s e (T ) = k. Case 2: Let e = xy where either x or y is an end vertex of T. Suppose x is an end vertex of T. Then by Corollary 2.9(ii), s e (T ) k 1. Let W = {u 1, u 2,...,u k } be the set of all end vertices of T. Then x W. Now let W = W {x}. Then W is an edge fixed Steiner set of G. Therefore s e (T ) k 1. Thus s e (T ) = k 1. Similarly if y is an end vertex of T, then also s e (T ) = k 1.

776 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi Theorem 2.11. Let e = xy be any edge of a connected graph G of order at least 3. Let v be a cut-vertex of G and W be an edge fixed Steiner set of G. (i) If v = x or y, then every component of G v contains an element of W. (ii) If v x and v y, then for each component C of G v with x, y / C, W C Φ. Proof: (i) Let v = x or y. Suppose there exists a component say G 1 of G v such that G 1 contains no vertex of W. By Theorem 2.8, W contains all the extreme vertices of G and hence G 1 does not have any extreme vertex of G. Thus G 1 contains at least one edge say uw distinct from xy. Since every edge fixed Steiner W-tree T must have its end vertex in W and v is a cut vertex of G, it is clear that no edge fixed Steiner W-tree would contain the edge uw. It contradicts that W is an edge fixed Steiner set of G. (ii) Let v x and v y. Suppose there exists a component say C of G v with x, y / C, W C = Φ. Then proceeding as in (i), we arrive at a contradiction. Corollary 2.12. If e is a cut-edge of a connected graph G and W is an edge fixed Steiner set of G, then e lies on a Steiner W- tree of G. Corollary 2.13. If v is a cut-vertex of a connected graph G and W is an edge fixed Steiner set of G, then v lies on a Steiner W- tree of G. Theorem 2.14. No cut - vertex of a connected graph G belongs to any minimum edge fixed Steiner set of G. Proof: Let v be a cut-vertex of G and W be a minimum edge fixed Steiner set of G. We claim that v / W. Suppose that v W. Let G 1, G 2, G 3,... G r (r 2) be the components of G v. By Theorem 2.11, each component G i (1 i r) contains an element of W. Let W = W {v}. Then by Corollary 2.13, each edge fixed Steiner W- tree contains v since v is a cut vertex. Now, since v / W ; it follows that each edge fixed Steiner W- tree is also an edge fixed Steiner W - tree of G. Thus W is also an edge fixed Steiner set of G such that W < W. This is a contradiction since W is a minimum edge fixed Steiner set of G. Theorem 2.15. For any edge e in the complete bipartite graph K m,n with m n and m + n 3, s e (K m,n ) = m + n 2.

Edge fixed Steiner number of a graph 777 Proof: Let U = {u 1, u 2,...,u m } and V = {v 1, v 2,...,v n } be the bipartition of K m,n. Let e = u i v j, 1 i m, 1 j n be an edge in K m,n. Let W = V {u i, v j }. Then clearly W is an edge fixed Steiner set of K m,n so that s e (K m,n ) m + n 2. Let X be a subset of the vertex set of K m,n such that X m + n 2. If X U V then X is connected. Hence X is not an s e -set of G. Thus s e (K m,n ) = m + n 2. Theorem { 2.16. For any edge e in the wheel W p = K 1 + C p 1 (p 6), p 5 if e is an edge in Cp 1 s e (W p )= p 2 otherwise Proof: Let {w, u 0, u 1, u 2,...,u p 2 } be the vertex set of W p where w K 1 and u 0, u 1, u 2,...,u p 2 C p 1. For instance, the wheel W 7 is as shown in Figure 2. u 0 u 1 u 5 w u 2 u 4 u 3 Figure 2: W 7 Case 1: Let e = u i u i+1 be an edge in C p 1 where i = 0, 1, 2,..., p 2. Let W = {u (i+3)mod(p 1), u (i+4)mod(p 1),...,u (i+p 3)mod(p 1) } and let W = W {u i, u i+1 }. Then the path T 1 : u imod(p 1), u (i+1)mod(p 1), u (i+2)mod(p 1), u (i+3)mod(p 1),..., u (i+p 4)mod(p 1),u (i+p 3)mod(p 1) is a Steiner W -tree. Similarly the path T 2 : u (i+1)mod(p 1), u imod(p 1), u (i+p 2)mod(p 1), u (i+p 3)mod(p 1),...,u (i+4)mod(p 1), u (i+3)mod(p 1) is another Steiner W -tree. It is clear that G = W p - {u (i+2)mod(p 1), u (i+p 2)mod(p 1) } is also connected. Then any spanning tree T 3 of G is necessarily a Steiner W -tree. So T 1, T 2 and T 3 are the Steiner W -trees containing W. Also W is the unique Steiner set of W p and hence W is the unique edge fixed Steiner set of W p. So in this case, s e (W p ) = p 5. Case 2: Let e be an edge not in C p 1. Then e = wu i where i = 0, 1, 2,... p 2. Let W = {u 0, u 1, u 2,..., u i 1, u i+1,..., u p 2 } and W = W {w, u i } = V(W p ). Then the star centered at w is the Steiner W -tree containing W. So W is the unique edge fixed Steiner set of W p. Hence in this case, s e (W p ) = p 2.

778 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi Theorem 2.17. For any edge e in the n - cube Q n (n 2), s e (Q n )= 2. Proof: For n = 4, the hypercube Q 4 is represented as shown in Figure 3. 1111 0111 1011 1101 1110 0011 0101 0110 1001 1010 1100 0001 0010 0100 1000 0000 Figure 3: Q 4 Let e = xy be any edge in Q n which contains 2 n vertices. Then x and y are of the form a 1 a 2 a 3...a n where a i {0, 1}. Since x and y are adjacent, they differ by 0 and 1 in only one position. Let W = {x c, y c } and W = W {x, y}. Since x and y differ by 0 and 1 in exactly one position, x c and y c also differ by 0 and 1 in exactly one position. So they are adjacent in Q n. Then any x-y c path containing y and x c is a Steiner W -tree. Similarly, any y-x c path containing x and y c is also a Steiner W -tree. It is possible to find such Steiner W -trees which cover all the vertices of Q n. Also W is the unique Steiner set of Q n and W is the unique edge fixed Steiner set of Q n. So s e (Q n ) = 2. Theorem 2.18. Let G be a connected graph. For any edge e = uv in G, s e (G) = 1 if and only if there exists a vertex y such that every vertex of G is on a diametral path joining either u and y or a diametral path joining v and y. Proof: Let e = uv be any edge in G. Let y be any vertex distinct from u and v of G such that every vertex of G is on a diametral path P joining either u and y or a diametral path joining v and y. Let W = {y}. We know that s e (G) 1. Since P is a diametral path joining either u and y or a diametral path joining v and y, it is clear that each

Edge fixed Steiner number of a graph 779 vertex of G is on a geodesic joining either u and y or a geodesic joining v and y and also every u-y geodesic or v-y geodesic is an edge fixed Steiner tree of G. Therefore W is a edge fixed Steiner set of G. Thus s e (G) 1. Hence s e (G) = 1. Conversely, let s e (G) = 1. Let W = {y} be a minimum edge fixed Steiner set of G. Since every edge fixed Steiner tree in G is either u-y geodesic or v-y geodesic, each vertex of G also lies on either u-y geodesic or v-y geodesic. We claim that d(u,y) = d(g) or d(v,y)= d(g). Suppose that d(u,y) < d(g). Then let w and x be any two vertices of G such that d(w,x) = d(g). Now it follows that w and x lie on distinct geodesics joining u and y. Hence d(u,y) = d(u,w)+d(w,y)...(1). And d(u,y)= d(u,x)+d(x,y)...(2). By the triangle inequality, d(w,x) d(w,u)+d(u,x)...(3). Since d(u,y) < d(w,x) d(w,u)+d(u,x) by (3). That is, d(u,y) < d(w,u)+d(u,x)...(4). Using (4) in (1), we have d(w,y) < d(w,u)+d(u,x) d(u,w) = d(u,x). Thus d(w,y)< d(u,x)...(5). Also by the triangle inequality, we have d(w,x) d(w,y)+d(y,x)...(6). Using (5) and (2) in (6),we have d(w,x) < d(u,x)+d(u,y) d(u,x) = d(u,y). Thus d(g) < d(u,y) which is a contradiction. Hence d(u,y) = d(g). Similarly, we have d(v,y) = d(g). Therefore either d(u,y) = d(g) or d(v,y) = d(g). Thus each vertex of G is on a diametral path joining either u and y or a diametral path joining v and y. Theorem 2.19. If G is a connected graph of order p, then s e (G) < p 2 if and only if there exists an edge fixed Steiner set W such that W {u, v} is disconnected for any edge e = uv in G. Proof: Let e = uv be any edge in G. Assume that s e (G) < p 2. Let W be an edge fixed Steiner set of G. We claim that W {u, v} is disconnected. Suppose that W {u, v} is connected. Then the edge fixed Steiner tree of G contains the members of W {u,v} only, which is a contradiction to W is an edge fixed Steiner set of G. So W {u, v} is disconnected. Conversely, let e = uv be any edge of G. Let W be an edge fixed Steiner set of G such that W {x, y} is disconnected. We claim that s e (G) < p 2. Suppose that s e (G) = p 2. Then G is complete and W = V (G) {u,v} is the unique edge fixed Steiner set of G such that W {u, v} is connected. This is a contradiction. So s e (G) < p 2.

780 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi 3 Edge fixed Steiner number and diameter of a graph We know that if G is a connected graph of order p 3, then 1 s e (G) p 2 for any edge uv in G. Also we have for an edge uv in G, s e (G) = 1 if and only if there exists a vertex y such that every vertex of G lies on either a diametral path joining u and y or a diametral path joining v and y. Here we give a strong upper bound for the edge fixed Steiner number of a graph by using its order and diameter. Theorem 3.1. For any edge e = xy in a nontrivial tree, s e (G) = p d or p d+1 if and only if T is a caterpillar. Proof: Let T be a non trivial tree. Let P: v 0, v 1, v 2,...,v d = v be a diametral path. Let k be the number of end vertices of T and l be the number of internal vertices of T other then v 1, v 2,...,v d 1. Then d 1 + l + k = p. By Corollary 2.13, s e (G) = k or k 1 for any edge xy in T. Hence s e (G) = p d l + 1 or p d l for any edge e in T. It implies that s e (G) = p d + 1 or p d if and only if l = 0 (i.e)if and only if all the internal vertices of T lie on a diametral path P (i.e)if and only if T is a caterpillar. For every connected graph G, rad G diam G 2 rad(g). Ostrand [7] proved that every two positive integers a and b with a b 2a are realizable as the radius and diameter, respectively, of some connected graph. It is extended for the edge fixed Steiner number also as follows. Theorem 3.2. For positive integers r, d and n 2 with r d 2r, there exists a connected graph G with rad G = r, diam G = d and s e (G) = n or n 1 for any edge e = xy in G. Proof: Let e = xy be any edge in G. When r = 1, we have d = 1 or 2. If d = 1, let G = K n+2. Then by Theorem 2.15, s e (G) = n for any edge e = xy in G. If d = 2, let G = K 1,n+1. By the Theorem 2.15, s e (G) = n for any edge e = xy in G. Let r 2. We construct a graph G with required properties as follows. Case 1: Let r = d. For n = 2, let G = C 2r. Then we have done s e (G) = 2 for all e in G. Now, let n 3. Let C 2r : { u 1, u 2,...,u 2r, u 1 } be a cycle of order 2r. Let G be the graph obtained by adding new vertices { x 1, x 2,...,x n 2 } and

Edge fixed Steiner number of a graph 781 joining each x i ( 1 i n 2 ) with u 1 and u 2 of C 2r. Then the graph G thus obtained is shown in Figure 4. u 2r 1 u 2r u 4 u 1 x n 3 x n 2 C 2r x 1 x 2 u 3 u 2 Figure 4: G It is seen that the eccentricity of each vertex of G is r so that rad G = diam G = r. Also S = { x 1, x 2,...,x n 1 } is the set of all extreme vertices of G with S = n 1. Let U = { u 1, u 2,...,u 2r }. Let e = xy where x, y U. By Theorem 2.8, S is contained in every edge fixed Steiner set of G. It is clear that S is not an edge fixed Steiner set of G and so s e (G) n. Let u and v be the antipodal vertices of x and y respectively in C 2r. Then it follows from Theorem 2.8 that S {u, v} is an edge fixed Steiner set of G so that s e (G) = n. Let e = u 1 x i or u 2 x i where x i S for i = 1, 2,...,n 1. Then by Theorem 2.8, S {x i } for i = 1, 2,...,n 1 is contained in every edge fixed Steiner set of G. It is clear that S {x i } is not an edge fixed Steiner set of G. Let u V(C 2r ) where u u 1, u 2. Then S {x i } {u} is not an edge fixed Steiner set of G and so s e (G) n. Let w and z be two antipodal vertices of x i. Then W = S {x i } {w,z} is an edge fixed Steiner set of G and hence s e (G) = n. Case 2: Suppose r < d. Let C 2r : (v 1, v 2,...,v 2r, v 1 ) be a cycle of order 2r and let P d r+1 : u 0, u 1,...,u d r be the path of order d r + 1. Let H be the graph obtained from C 2r and P d r+1 by identifying v 1 in C 2r and u 0 in P d r+1. If n = 2, then let G = H. Let e = xy be any edge in G. Clearly s e (G) = 1 or 2 according as xy {v r v r+1, v r+1 v r+2, u d r 1 u d r } or xy {v 1 u 1, u 1 u 2, u 2 u 3,...,u d r 2 u d r 1, v 1 v 2, v 2 v 3,...,v r 1 v r, v r+2 v r+3,..., v 2r v 1 }. If n 3, then add (n 2) new vertices w 1, w 2,...,w n 2 to H and joining each vertex w i ( 1 i n 2 ) to the vertex u d r 1 and obtain the graph G, which is shown in Figure 5. Now rad G = r and diam G = d and G has n 1 end vertices.

782 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi v r+2 v 2r v r+1 vr u 1 u 2 u d r 1 u d r C 2r v 1 = u 0 w n 2 v w 1 2 w 2 Figure 5: G Clearly s e (G) = n or n 1 according as xy {v 1 u 1, u 1 u 2,..., u d r 2 u d r 1, v 1 v 2, v 2 v 3,..., v r 1 v r, v r+2 v r+3,..., v 2r v 1 } or xy {v r v r+1 4, v r+1 v r+2, u d r 1 u d r, u d r w 1, u d r w 2,..., u d r w n 2 }. 4 The Steiner number and edge fixed Steiner number of a graph In this section, we establish the equivalence of minimum Steiner set and minimum edge fixed Steiner set in a connected graph with cut edges. We determine the relation between the Steiner number and the edge fixed Steiner number of a graph. Also for any two positive integers a and b with 2 a b, we construct a connected graph G with s(g) = a and s e (G) = b for some edge e = xy in G. Theorem 4.1. Let G be a connected graph with cut edges. Let e = xy be a cut edge in G. Then W V(G) is an s e -set of G if and only if W is an s-set of G. Proof: Let e = xy be a cut edge in G. Let W be an s e -set of G. If W is not an s-set of G, then there exists a set W with W < W such that W is a minimum Steiner set of G. Since e is a cut edge of G, both x and y lie on every edge fixed Steiner W -tree of G. Hence it follows that W is an s e -set of G which leads to a contradiction. Therefore W is an s-set of G. Conversely, Let W be an s-set of G. Since e = xy is a cut edge of G, both x and y lie in every Steiner W tree of G. Hence it is clear that W is an edge fixed Steiner set of G. Suppose that W is not an s e -set of G. Then there exists a set W with W < W such that W is an s e -set of G. Hence every vertex of G lies on an edge fixed Steiner W tree of G. Also since x, y belong to every edge fixed Steiner W tree of G, every edge fixed Steiner W tree of G is also a Steiner W tree of G. So it follows that W is a Steiner set of G which is a contradiction. Hence W is an s e -set of G.

Edge fixed Steiner number of a graph 783 Corollary 4.2. Let e = xy be a cut edge of a connected graph G. Then s e (G) = s(g). The corollary immediately follows from Theorem 4.1. Remark 4.3. The converse of the Corollary 4.2 need not be true. For example, consider the graph G shown in Figure 1. As illustrated in Example 2.2, W = {v 1, v 4 } is the minimum Steiner set of G so that s(g) = 2. Also for the edges v 2 v 6 and v 3 v 5, W = {v 1, v 4 } is the minimum edge fixed Steiner set of G. Hence s e (G) = 2 = s(g). But v 2 v 6 and v 3 v 5 are not cut edges of G. Theorem 4.4. For any edge e = xy in G, s(g) s e (G) + 2. Proof: Let e = xy be any edge in G and W be an edge fixed Steiner set of G. Then W {x,y} is a Steiner set of G so that s(g) W {x, y} = s e (G) + 2. Theorem 4.5. For any edge e = xy in G, s(g) = s e (G) + 2 if and only if the x and y belong to a minimum Steiner set of G. Proof: Let W be a minimum Steiner set of G and e = xy be an edge in G such that x, y W. Then W {x,y} is an edge fixed Steiner set of G so that s e (G) W {x, y} = s(g) 2. It implies that s e (G) + 2 s(g). From Theorem 4.4, we have s e (G) + 2 s(g). Therefore s(g) = s e (G) + 2. Conversely, let s(g) = s e (G) + 2 for any edge e = xy in G. Let W be a minimum edge fixed Steiner set of G. Then W = W {x,y} is a Steiner set of G. Since cardinality of W = s e (G) + 2. it follows that W is a minimum Steiner set of G with x, y W. Hence the theorem. Theorem 4.6. For any two positive integers a and b with 2 a b, there exists a connected graph G with s(g) = a and s e (G) = b for some edge e = xy in G. Proof: Case 1: Let a = b. When a = 2, for any edge e in an even cycle G, we have s e (G) = 2 by Theorem 2.6 and s(g) = 2 by Theorem 1.3. Now a = b with b 3 and 3 a = b. Let G be a tree with a end vertices. Then by theorem 1.2, s(g) = a. Let e = xy be an intermediate vertex of G. Then by Corollary 2.13 s e (G) = a. Case 2: 3 < a < b.

784 M. Perumalsamy, P. Arul Paul Sudhahar, J. John and R. Vasanthi Let G be the graph obtained in Figure 6 from the path P on the vertices: u 1, u 2, u 3, by adding the new vertices v 1, v 2,...,v b a+2 and w 1, w 2,...,w a 2 and joining each v i (1 i b - a + 2) with u 1 and u 3 and also joining each w i (1 i a-2) with u 1 and u 2. Let W = {w 1, w 2,...,w a 2 }. Then W is the set of extreme vertices of G. w a 2 w 2 w 1 u 1 u 2 u 3 v 1 v 2 v 3 v b a+1 Figure 6: Let S be any Steiner set of G. Then by Theorem 1.1, W S. It is clear that W is not a Steiner set of G. Also it is verified that W = W {v}, where v/ W is not a Steiner set of G. But, it is seen that W {u 1, u 3 } is a Steiner set of G and so s(g) = a. Next, we prove that s e (G) = b for some e G. Let e = u 3 v 2. By Theorem 2.8, W is a subset of every s e set of G. It is clear that W is not an edge fixed Steiner set of G. Also, it is easily seen that every edge fixed Steiner set contains each v i (1 i b - a + 1) except the vertices u 3, v 2 and so s e (G) a-2+b-a+1 = b 1. It is easily verified that Z = W {v 1, v 2,...,v b a+1 } (except the vertices u 3, v i ) is not an s e set of G and so s e (G) b. Now Z = Z {u 1 } is an s e set of G so that s e (G) = b. References [1] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, 1990.

Edge fixed Steiner number of a graph 785 [2] G. Chartrand F. Harary and P. Zang, On the geodetic number of a graph, Networks, 39 (2002), 1-6. https://doi.org/10.1002/net.10007 [3] G. Chartrand and P. Zhang, The Steiner number of a graph, Discrete Mathematics, 242 (2002) 41-54. https://doi.org/10.1016/s0012-365x(00)00456-8 [4] Gary Chatrand and Ping Zhang,Introduction to Graph Theory, Tata Mc- Graw Hill Education Private Limited, New Delhi, Eighth Reprint, 2012. [5] F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969. [6] F. Harary, E. Loukakis, C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling, 17 (1993), no. 11, 87-95. https://doi.org/10.1016/0895-7177(93)90259-2 [7] P.A. Ostrand, Graphs with specified radius and diameter, Discrete Math., 4 (1973), 71-75. https://doi.org/10.1016/0012-365x(73)90116-7 [8] A.P. Santhakumaran, P. Titus, Vertex Geodomination in Graphs, Bulletin of Kerala Mathematics Association, 2 (2005), no. 2, 45-57. [9] A. P. Santhakumaran and J. John, The edge Steiner number of a graph, Journal of Discrete Mathematical Science and Cryptography, 10 (2007), no. 5, 677-696. https://doi.org/10.1080/09720529.2007.10698149 [10] A.P. Santhakumaran and P. Titus, The edge fixed geodomination number of a graph, An. St. Univ. Ovidius Constanta, 17 (2009), no. 1, 187-200. Received: July 14, 2017; Published: August 11, 2017