The Restrained Edge Geodetic Number of a Graph

Transcription

1 International Journal of Computational and Applied Mathematics. ISSN Volume 11, Number 1 (2016), pp Research India Publications The Restrained Edge Geodetic Number of a Graph A. P. Santhakumaran and M. Mahendran Department of Mathematics, Hindustan University, Hindustan Institute of Technology and Science, Chennai , India. apskumar1953@gmail.com, magimani83@gmail.com P. Titus Department of Mathematics, University College of Engineering Nagercoil, Anna University, Tirunelveli Region, Nagercoil , India. titusvino@yahoo.com Abstract A set S of vertices of a connected graph G is a geodetic set if every vertex of G lies on an x y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is the geodetic number of G, denoted by g(g). A set S of vertices of a graph G is an edge geodetic set if every edge of G lies on an x y geodesic for some elements x and y in S. The minimum cardinality of an edge geodetic set of G is the edge geodetic number of G, denoted by eg(g). A set S of vertices of a graph G is a restrained edge geodetic set if S is an edge geodetic set, and if either V = S or the subgraph G[V S] induced by V S has no isolated vertices. The minimum cardinality of a restrained edge geodetic set of G is the restrained edge geodetic number of G and is denoted by eg r (G). The restrained edge geodetic numbers of some standard graphs are determined. Some special classes of graphs of order p with eg r (G) = p are characterized. It is proved that, for the integers a,b,c and p such that 2 a b c p with p c a 2 0, there exists a connected graph G of order p, geodetic number a, edge geodetic number b and the restrained edge geodetic number c except the values (a, b) {(2, 2), (2, 3), (3, 4)}. It is also proved that If a,b,c and p are integers such that 3 a b c p 3, then there exists a connected graph G of order p, geodetic number a, restrained geodetic number b and the restrained edge geodetic number c. AMS subject classification: 05C12. Keywords: geodetic set, geodetic number, edge geodetic set, edge geodetic number, restrained edge geodetic set, restrained edge geodetic number.

2 10 A. P. Santhakumaran, M. Mahendran and P. Titus 1. Introduction By a graph G = (V, E) we mean a simple connected graph of order at least two. The order and size of G are denoted by p and q, respectively. For basic graph theoretic terminology, we refer to Harary [5]. The neighborhood of a vertex v is the set N(v) consisting of all vertices u which are adjacent with v. The closed neighborhood of a vertex v is the set N[v] =N(v) {v}. A vertex v is an extreme vertex if the subgraph induced by its neighbors is complete. For a cut-vertex v in a connected graph G and a component H of G v, the subgraph H and the vertex v together with all edges of G joining v to V(H)is called a branch of G at v. The eccentricity e(v) of a vertex v in G is e(v) = max{d(v,u) : u V (G)}. The radius, rad G of G is rad G = min{e(v) : v V (G)} and the diameter, diam G of G is diam G = max{e(v) : v V (G)}. For any two vertices x and y in a connected graph G, the distance d(x,y) is the length of a shortest x y path in G. An x y path of length d(x,y) is called an x y geodesic.a vertex v is said to lie on an x y geodesic P if v is a vertex of P including the vertices x and y. A graph G is geodetic if every pair of vertices in G is joined by a unique geodesic. The closed interval I[x,y] consists of all vertices lying on some x y geodesic of G, while for S V, I[S] = I[x,y]. A set S of vertices is a geodetic set if x,y S I[S] =V,and the minimum cardinality of a geodetic set is the geodetic number g(g). A geodetic set of cardinality g(g) is called a g-set. The geodetic number of a graph was introduced in [1,6] and further studied in [2,3,5]. A set S of vertices of a graph G is an edge geodetic set if every edge of G lies on an x y geodesic for some elements x and y in S. The minimum cardinality of an edge geodetic set of G is the edge geodetic number of G, denoted by eg(g). The edge geodetic number was introduced and studied in [8]. These concepts have interesting applications to the problem of designing the route for a shuttle and communication network design. The following theorems will be used in the sequel. Theorem 1.1. [5] Let v be a vertex of a connected graph G. The following statements are equivalent: (i) v is a cut vertex of G. (ii) There exist vertices u and w distinct from v such that v is on every u w path. (iii) There exists a partition of the set of vertices V {v} into subsets U and W such that for any vertices u U and w W, the vertex v is on every u w path. Theorem 1.2. [8] Each extreme vertex of a connected graph G belongs to every edge geodetic set of G. Theorem 1.3. [8]If G has exactly one vertex v of degree p 1, then eg(g) = p 1. Throughout this paper G denotes a connected graph with at least two vertices.

3 The Restrained Edge Geodetic Number of a Graph Restrained Edge Geodetic Number of a Graph Definition 2.1. A set S of vertices of a graph G is a restrained edge geodetic set if S is an edge geodetic set, and if either S = V or the subgraph G[V S] induced by V S has no isolated vertices. The minimum cardinality of a restrained edge geodetic set of G is the restrained edge geodetic number of G, and is denoted by eg r (G). Example 2.2. For the graph G given in Figure 2.1, it is clear that S ={u, w} is the minimum geodetic set of G and so g(g) = 2; S 1 = {u, v, w, x} is the minimum edge geodetic set of G and so eg(g) = 4; and S 2 ={u,v,w,x,y,z} is the minimum restrained edge geodetic set of G and so eg r (G) = 6. Thus the geodetic number, the edge geodetic number and the restrained edge geodetic number of a graph are all different. Every restrained edge geodetic set is an edge geodetic set, and the converse is not true. For the graph G given in Figure 2.1, S 1 is an edge geodetic set, however it is not a restrained edge geodetic set of G. Also, every edge geodetic set is a geodetic set and so every restrained edge geodetic set is a geodetic set of a connected graph G. It was shown in [6] that determining the geodetic number of a graph is an NP-hard. Hence determining the restrained edge geodetic number of a graph is also an NP-hard problem. Since every restrained edge geodetic set of G is an edge geodetic set, the next result follows from Theorem 1.3. Theorem 2.3. Each extreme vertex of a connected graph G belongs to every restrained edge geodetic set of G. Corollary 2.4. or the complete graph K p (p 2), eg r (K p ) = p. Theorem 2.5. For any connected graph G, 2 g(g) eg(g) eg r (G) p. Remark 2.6. If eg(g) = p or p 1, then eg r (G) = p. The converse is not true. For the cycle C 5, eg(c 5 ) = 3 = p 2 and eg r (C 5 ) = 5 = p.

4 12 A. P. Santhakumaran, M. Mahendran and P. Titus Theorem 2.7. There is no graph G of order p with eg r (G) = p 1. Proof. Since every restrained edge geodetic set of G is an edge geodetic set of G and the complement of each restrained edge geodetic set has cardinality different from 1, we have eg r (G) = p 1. Theorem 2.8. If G has exactly one vertex v of degree p 1, then eg r (G) = p. Proof. The result follows from Theorems 1.3, 2.5 and 2.7. The converse of Theorem 2.8 need not be true since for the complete graph K p, eg r (K p ) = p. Corollary 2.9. For any tree T with p 3 vertices, eg r (T ) = p if and only if T is a star. Corollary For any tree T with p 4 vertices, eg r (T ) = p 2 if and only if T is a double star. The following theorem is easy to verify. Theorem (i) If T is a tree with k end vertices, then { p if T is a star eg r (T ) = k if T is not a star. (ii) For the cycle C p (p 3), p for p {3, 4, 5} eg r (C p ) = 2 for p 6 and p is even 3 for p 7 and p is odd. (iii) For the wheel W p = K 1 + C p 1 (p 5), eg r (W p ) = p. (iv) For the complete bipartite graph K m,n (m, n 2), eg r (K m,n ) = min{m, n}+1. (v) For the hyper cube Q p, eg r (Q p ) = 2. Theorem Let G be a connected graph with every vertex of G is either a cut vertex or an extreme vertex. Then eg r (G) = p if and only if G = K 1 + m j K j. Proof. Let G = K 1 + m j K j. Then G has at most one cut vertex. Suppose that G has no cut vertex. Then G = K p, hence by Corollary 2.4, eg r (G) = p. Suppose that G has exactly one cut vertex. Then all the remaining vertices are extreme and hence by Theorem 2.3, eg r (G) = p.

5 The Restrained Edge Geodetic Number of a Graph 13 Conversely, suppose that eg r (G) = p. If p = 2, then G = K 2 = K 1 + K 1. If p 3, then there exists a vertex x, which is not a cut vertex of G. If G has two or more cut vertices, then the induced subgraph of the cut vertices is a non-trivial path. Then the set of all extreme vertices is a minimum restrained edge geodetic set of G and so eg ( G) p 2, which is a contradiction. Thus the number of cut vertices k of G is at most one. Case 1. If k = 0, then the graph G is a block. If p = 3, then G = K 3 = K 1 + K 2.If p 4, we claim that G is complete. Suppose that G is not complete. Then there exist two vertices x and y in G such that d(x,y) 2. By Theorem 1.1, both x and y lie on a common cycle and hence x and y lie on a smallest cycle C : x,x 1,...,y,...,x n,x of length at least 4. Thus every vertex of C on G is neither a cut vertex nor an extreme vertex, which is a contradiction to the assumption. Hence G is the complete graph K p and so G = K 1 + K p 1. Case 2. If k = 1, let x be the cut vertex of G. If p = 3, then G = P 3 = K 1 + m j K 1, where m j = 2. If p 4, we claim that G = K 1 + m j K j, where m j 2. It is enough to prove that every block of G is complete. Suppose that there exists a block B, which is not complete. Let u and v be two vertices in B such that d(u, v) 2. Then by Theorem 1.1, both u and v lie on a common cycle and hence u and v lie on a smallest cycle of length at least 4. Hence every vertex of C on G is neither a cut vertex nor an extreme vertex, which is a contradiction. Thus every block of G is complete so that G = K 1 + m j K j, where K 1 is the vertex x and m j The Restrained Edge Geodetic Number and Diameter of a Graph We have seen that if G is a connected graph of order p 2, then 2 eg r (G) p. In the following theorem we give an improved upper bound for the restrained edge geodetic number of a geodetic graph in terms of its order and diameter. Theorem 3.1. If G is a geodetic graph of order p and diameter d = 2, then eg r (G) p d + 1. Proof. If d = 1, then G = K p, so that eg r (G) = p = p d + 1. So, let d 3. Let u and v be two vertices of G such that d(u, v) = d and let u = v 0,v 1,...,v d = v be a u v geodesic of length d. Clearly, S = V (G) {v 1,v 2,...,v d 1 } is a restrained edge geodetic set of G and hence eg r (G) p d + 1. It is observed that for the path P 3, d = 2 and eg r (P 3 ) = 3 = p>p d + 1. Hence Theorem 3.1 is not true for d = 2. Also, if G is not a geodetic graph, then Theorem 3.1 is not true. For the graph G given in Figure 3.1, eg r (G) = 6, p= 6, and d = 3 so that eg r (G)>p d + 1.

6 14 A. P. Santhakumaran, M. Mahendran and P. Titus Remark 3.2. The converse of Theorem 3.1 need not be true. For the graph G given in Figure 3.2, p = 7, d= 4, eg r (G) = 3 and p d + 1 = 4 so that eg r (G) p d + 1. However, G is not a geodetic graph. A caterpillar is a tree for which the removal of all the end vertices gives a path. Theorem 3.3. For every non-trivial tree T with diameter d 3, eg r (T ) = p d + 1if and only if T is a caterpillar. Proof. Let T be any non-trivial tree. Let P : u = v 0,v 1,...,v d = v be a diametral path. Let k be the number of end vertices of T and let l be the number of internal vertices of T other than v 1,v 2,...,v d 1. Then d 1 + l + k = p. By Corollary 2.12(i), eg r (T ) = k and so eg r (T ) = p d l + 1. Hence eg r (T ) = p d + 1 if and only if l = 0, if and only if all the internal vertices of T lie on the diametral path P, if and only if T is a caterpillar. For every connected graph G, rad G diam G 2 rad G. Ostrand [7] showed that every two positive integers a and b with a b 2a are realizable as the radius and diameter, respectively, of some connected graph. Ostrand s theorem can be extended so that the restrained edge geodetic number can be prescribed. Theorem 3.4. For the positive integers r, d and n 4 with r d 2r, there exists a connected graph G with rad G = r, diam G = d and eg r (G) = n. Proof. If r = 1, then d = 1or2. Ifd = 1, then G = K n has the desired properties.

7 The Restrained Edge Geodetic Number of a Graph 15 If d = 2, then G = K 2 + (K n 1 K 1 ) has the desired properties. If r = d = 2, then G = K 2,n 2 has the desired properties. Now, let r 3. We construct a graph G with the desired properties as follows: Case 1. r = d. Let G be the graph obtained from disjoint union of a cycle C 2r : v 1,v 2,...,v 2r,v 1 and the complete graph K n 2 by joining v 1 and v 2r to all vertices of K n 2. It is clear that rad G = diam G = r and V(K n 2 ) {v 1,v r+1 } is a minimum restrained edge geodetic set of G. Hence eg r (G) = n. Case 2. r < d. Let C 2r : v 1,v 2,...,v 2r,v 1 be a cycle of order 2r and and let P d r+1 : u 0,u 1,u 2,...,u d r be a path of order d r + 1. Let H be the graph obtained from C 2r and P d r+1 by identifying v 1 and u 0. Now, add n 2 vertices w 1,w 2,...,w n 2 to H by joining each vertex w i (1 i n 2) to the vertex u d r 1 and obtain the graph G given in Figure 3.3. It is clear that rad G = r, diam G = d and {w 1,w 2,...,w n 2,u d r,v r+1 } is a minimum restrained edge geodetic set of G. Hence eg r (G) = n. Theorem 3.5. If p, d and n are integers such that 3 d p 1, 2 n p and p d n + 1 0, then there exists a graph G of order p, diameter d and eg r (G) = n. Proof. If n = 2, let P d+1 : u 0,u 1,u 2,...,u d be a path of length d. Add p d 1 new vertices w 1,w 2,...,w p d 1 to P d+1 and join these to both u 0 and u 2, there by producing the graph G of Figure 3.4. Then G has order p and diameter d. Clearly, S ={u 0,u d } is the minimum restrained edge geodetic set of G and so eg r (G) = 2 = n.

8 16 A. P. Santhakumaran, M. Mahendran and P. Titus If 3 n p, then add n 2 new vertices v 1,v 2,...,v n 2 to P d+1 and join these to u 1, also add p d n + 1 new vertices w 1,w 2,...,w p d n+1 to P d+1 and join these to both u 0 and u 2, there by producing the graph G of Figure 3.5. Then G has order p and diameter d. Clearly, S ={v 1,v 2,...,v n 2,u 0,u d } is the minimum restrained edge geodetic set of G and so eg r (G) = n. Theorem 3.6. Let a,b,c and p be integers such that 2 a b c p with p c a 2 0. Then there exists a connected graph G of order p, geodetic number a, edge geodetic number b and the restrained edge geodetic number c except the values (a, b) {(2, 2), (2, 3), (3, 4)}. Proof. Case 1. 4 a b c. Let G 1 = K 2,c b+2 be the complete bipartite graph with bipartite sets X ={x 1,x 2 } and Y ={y 1,y 2,...,y c b+2 }, G 2 = K 1,b with bipartite sets {v} and {v 1,v 2,w 1,w 2,...,w a 4,z 1,z 2,...,z b a+2 }, and G 3 = P : u 1,u 2,...,u p c 5. Let G be the graph obtained from G 1,G 2 and G 3 by (i) joining the vertices x 1,x 2 of G 1 with the vertex u 1 of G 3, (ii) joining the vertex v of G 2 with the vertex u p c 5 of G 3, and (iii) joining the vertices z 1,z 2,...,z b a+2 with the vertices v 1 and v 2 in G 2. The graph G is shown in Figure 3.6 and its order is p.

9 The Restrained Edge Geodetic Number of a Graph 17 It is clear that S 1 ={x 1,x 2,v 1,v 2,w 1,w 2,...,w a 4 } is a minimum geodetic set of G, S 2 = S 1 {z 1,z 2,...,z b a+2 } is a minimum edge geodetic set of G and S 3 = (S 2 {x 1,x 2 }) {y 1,y 2,...,y c b+2 } is a minimum restrained edge geodetic set of G and so g(g) = a, eg(g) = b and eg r (G) = c. Case 2. 2 a<b c and b = a + 1. Let P : u 1,u 2,...,u p c a+4 be a path of order p c a + 4. Take a set S ={w 1,w 2,...,w a 1,y 1,y 2,...,y c b,z 1,z 2,...,z b a 1 } of new vertices. Let G be the graph obtained from P and S by (i) joining the vertices w 1,w 2,...,w a 1 with the vertex u p c a+4, (ii) joining the vertices y 1,y 2,...,y c b with the vertices u 1 and u 3, and (iii) joining the vertices z 1,z 2,...,z b a 1 with the vertices u 1,u 2 and u 3. The graph G is shown in Figure 3.7 and its order is p. It is easily verified that S 1 ={u 1,w 1,w 2,...,w a 1 } is a minimum geodetic set of G, S 2 = S 1 {u 2,z 1,z 2,...,z b a+1 } is a minimum edge geodetic set of G and S 3 = S 2 {y 1,y 2,...,y c b } is a minimum restrained edge geodetic set of G and so g(g) = a, eg(g) = b and eg r (G) = c. Case 3. 3 a = b c. Let G 1 = K 2,c b+2 be the complete bipartite graph with bipartite sets X ={x 1,x 2 } and Y ={y 1,y 2,...,y c b+2 }, let G 2 = K 1,a 2 be a star with bipartite sets {u} and {w 1,w 2,...,w a 2 }, and let G 3 : u 1,u 2,...,u p c 3 be a path of order p c 3. Let G be the graph obtained from G 1,G 2 and G 3 by joining the vertices x 1,x 2 of G 1 with the vertex u 1 of G 3 and joining the vertex v of G 2 with the vertex u p c 3 of G 3. The graph G is shown in Figure 3.8 and its order is p. It is easily verified that S 1 ={x 1,x 2,w 1,w 2,...,w a 2 } is both a minimum geodetic set and a minimum edge geodetic set of G and S 2 = (S 1 {x 1,x 2 }) {y 1,y 2,...,y c a+2 } is a minimum restrained edge geodetic set of G and so g(g) = eg(g) = a and eg r (G) = c. Theorem 3.7. For any connected graph G, 2 g(g) g r (G) eg r (G) p.

10 18 A. P. Santhakumaran, M. Mahendran and P. Titus Proof. Since every restrained edge geodetic set is a restrained geodetic set, the result follows. Theorem 3.8. If a,b,c and p are integers such that 3 a b c p 3, then there exists a connected graph G of order p, geodetic number a, restrained geodetic number b and the restrained edge geodetic number c. Proof. Let G 1 = K 2,b a+2 be the complete bipartite graph with partite sets X ={x 1,x 2 } and Y ={y 1,y 2,...,y b a+2 } and let G 2 = P : v 1,v 2,...,v p c be a path of order p c. Let H be the graph obtained from G 1 and G 2 by identifying the vertex x 2 in G 1 with the vertex v 1 in G 2. Add c b + a 3 new vertices {u 1,u 2,...,u a 3,w 1,w 2,...,w c b } to H and join each vertex u i (1 i a 3) with the vertex v 2 and join each vertex w i (1 i c b) with the vertices v 1,v 2 and v 3. The graph G is shown in Figure 3.9 and its order is p. It is easily verified that S 1 ={x 1,x 2,v p c,u 1,u 2,...,u a 3 } is a minimum geodetic set of G, S 2 = (S 1 {x 1,x 2 }) {y 1,y 2,...,y b a+2 } is a minimum restrained geodetic set of G and S 3 = S 2 {w 1,w 2,...,w c b } is a minimum restrained edge geodetic set of G and so g(g) = a, g r (G) = b and eg r (G) = c.

11 The Restrained Edge Geodetic Number of a Graph 19 References [1] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, (1990). [2] F. Buckley, F. Harary, and L. V. Quintas, Extremal Results on the Geodetic Number of a Graph, Scientia A2 (1988) [3] G. Chartrand, F. Harary, and P. Zhang, On the Geodetic Number of a Graph, Networks. 39 (2002) 1 6. [4] G. Chartrand, G.L. Johns, and P. Zhang, On the Detour Number and Geodetic Number of a Graph, Ars Combinatoria 72 (2004) [5] F. Harary, Graph Theory, Addison-Wesley, [6] F. Harary, E. Loukakis, and C. Tsouros, The Geodetic Number of a Graph, Math. Comput. Modeling 17(11)(1993) [7] P.A. Ostrand, Graphs with specified radius and diameter, Discrete Math. 4(1973) [8] A. P. Santhakumaran, Edge Geodetic Number of a Graph,J. Discrete Math. Sci. & Cryptography, Vol. 10(2007), No. 3, [9] A.P. Santhakumaran, P. Titus and K. Ganesamoorthy, Restrained monophonic number of a graph, Communicated. [10] P. Titus, A.P. Santhakumaran and K. Ganesamoorthy, Restrained edge monophonic number of a graph, Communicated.

12