On the Geodetic Number of Line Graph

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 7, 01, no. 46, On the Geodetic Number of Line Graph Venkanagouda M. Goudar Sri Gouthama Research Center [Affiliated to Kuvempu University] Department of Mathematics, Sri Siddhartha Institute of Technology Tumkur, Karnataka, India K. S. Ashalatha Sri Gouthama Research Center [Affiliated to Kuvempu University] Sri Siddhartha Institute of Technology, Tumkur, Karnataka, India Department of Mathematics, Government First Grade College, Gubbi, Tumkur, Karnataka, India Venkatesha Department of Mathematics, Kuvempu University Shankarghatta, Shimoga, Karnataka, India M. H. Muddebihal Department of Mathematics, Gulbarga University Gulbarga Karnataka, India Abstract For any graph,, the line graph, whose vertices corresponds to the edges of G and two vertices in are adjacent if and only if the corresponding edges in G are adjacent. For two vertices u and v of G the set, consists of all vertices lying on a u-v geodesic in G. If S is a set of vertices of G then is

2 90 Venkanagouda M. Goudar et al the union of all sets, for vertices u and v in S. The geodetic number is the minimum cardinality among the subsets S of with. In this paper we obtain the geodetic number of line graph of any graph. Also, obtain many bounds on geodetic number in terms of elements of G and covering number of G. Mathematics Subject Classification: 05C05, 05C1 Keywords. Cross product, Distance, Geodetic number, Line graph, Vertex covering number 1. Introduction In this paper we follow the notations of 6. All the graphs considered here are finite, non trivial, undirected and connected. As usual denote the number of vertices and edges of a graph G, respectively. For any graph,, the Line graph whose vertices correspond to the edges of G and two vertices in are adjacent if and only if the corresponding edges in G are adjacent. The distance, between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. It is well known that this distance is a metric on the vertex set. For a vertex v of G, the eccentricity is the distance between v and a vertex farthest from v. The minimum eccentricity among the vertices of G is the radius, rad (G), and the maximum eccentricity is its diameter, diam (G). A u-v path of length, is called a u-v geodesic. We define, to be the set (interval) of all vertices lying on some u-v geodesic of G and for a nonempty subset S of, I(S) = U u, v S I (u, v). A set S of vertices of G is called a geodetic set in G if and a geodetic set of minimum cardinality is a minimum geodetic set.the cardinality of a minimum geodetic set in G is called the geodetic number. Now we define geodetic number of line graph of a graph G. A set S of vertices of is called a geodetic set in H if and a geodetic set of minimum cardinality is the geodetic number of and is denoted by. The Cartesian product (or direct product) of two sets is the set of all possible ordered pairs whose first component is a member of and whose second component is a member of. A vertex v is an extreme vertex in a graph G, if the sub graph induced by its neighbors is complete. A vertex cover (edge cover) in a graph G is a set of vertices (edges) that covers all edges (vertices) of G. The

3 On the geodetic number of line graph 91 minimum number of vertices (edges) in a vertex cover (edge cover) of G is the vertex cover number α ( G) (edge cover number of G. 0 For any undefined terms in this paper, see5,6.. Preliminary Results Theorem.1[4] Every geodetic set of a graph contains its extreme vertices. Theorem.[4] If G is a non trivial connected graph of order n and diameter d, then1. Theorem.3[4] Let G be a connected graph of order at least 3. If G contains a minimum geodetic set S with a vertex x such that every vertex of G lies on some geodesic in G for some, then. Propositions 1 The end edges of a tree T are the extreme vertices of a line Graph of T. Proposition For any tree T with order n and diameter d, have the same value of. 3. Main Results. Theorem 3.1 For any tree T with k end edges,. Proof. Let S be the set of all extreme vertices of a line graph of a tree T. By the theorem.1,. On the other hand, for an internal vertex v of T, there exists, of T such that v lies on the unique, geodesic in T. The corresponding end edges of T are the extreme vertices of.thus. Also every geodesic set of must contain S which is the unique minimum geodesic set. Thus. Hence. Corollary For any path with n vertices,. Proof. Clearly the set of two end edges of a path is its unique geodesic set. From theorem 3.1, the result follows. Theorem 3. For any tree T of order n and diameter d, then 1. Proof. Let T be a non trivial connected graph of order n and diameter d, let u and v be vertices of, for which,. Let u =,,, be a u-v path of length d. Now, let,,. From the proposition, and consequently 1.

4 9 Venkanagouda M. Goudar et al, Theorem 3.3 For cycle of order 3, 3,. Proof. The line graph L( ) of a cycle is again a cycle. For the cycle,, the set of any two antipodal vertices is a geodetic set of. Also for,1, no two vertices form a geodetic set, since there exists a 3 vertex geodetic set. Thus, 3,. Theorem 3.4 If every non end vertex of a tree T is adjacent to at least one end vertex, then, where k is number of end vertices in T. Proof. If 3, then the result is obvious. Let 3 and,, be the set of all end vertices in T with. Now without loss of generality, every end edge of T are the extreme vertices of. Suppose does not contain any end vertex then,,, where, forms a geodetic set of. Further if contains at least one end vertex w, then the set forms a geodetic set. Therefore in all the cases, we obtain. Theorem 3.5 For any tree T, with m edges,, where is an edge covering number. Proof. Suppose,, be the set of all end edges in T. Then where, be the minimal set of edges which covers the vertices of T and is not covered by, such that. Now without loss generality in, let, be the set of all vertices in formed by the end edges in T is the minimal geodetic set of. Clearly it follows that. Theorem 3.6 Let be the graph obtained by adding an end edge to a cycle with then 3, if n is even. Proof. Let,, be a cycle with n vertices which is even and let be the graph obtained from by adding an end edge such that. By the definition of line graph, has as an induced sub graph. Also the edge becomes a vertex of and it belongs to some geodetic set of. Hence,, are the vertices of where, are the edges incident on the antipodal vertex of u in, and these vertices belongs to some geodetic set of.. Since S =,, is the minimum geodetic set. Therefore 3. Theorem 3.7 be the graph obtained by adding an end edge to a cycle with, then, if n is odd.

5 On the geodetic number of line graph 93 Proof. Let,, be a cycle with n vertices which is odd and let be the graph obtained from by adding an end edge such that. By the definition of line graph, has as an induced sub graph, also the edge becomes a vertex of. Let, such that,, in the graph. Two elements subset of, of has the property that. Thus. Theorem 3.8 Let be the graph obtained by adding end edge, 1, to each vertex of such that,. Then. Proof. Let,,,, be a cycle with n vertices and. Let be the graph obtained by adding end edge 1,, to each vertex of G such that,. Clearly n be the number of end vertices of. By the definition of line graph, have n copies of as an induced sub graph. The edges for all i, becomes n vertices of and those lies on geodetic set of. Since they forms the extreme vertices of, by theorem.1. Theorem 3.9 For any cycle, n is even,. Proof. Let 3, is even be number of vertices and is the vertex covering number of G. We have and by theorem 3.3,. Also for even cycle, vertex covering number. Hence. Theorem 3.10 For cycle,, n is odd, 1. Proof. Let 3, is odd be the number of vertices and is the vertex covering number of. We have and by theorem 3.3, 3. Also for an odd cycle, vertex covering number Hence Theorem 3.11 For any integers,,, 1. Proof. Let and be the number of vertices and edges of the given graph, and d be the diameter. Since diameter of,, the number of vertices in, is. Hence by theorem. 1. Now we have, 1., 1. Theorem 3.1 For any integer 4,. Proof. Let 4 be the vertices of the given graph and d be the diameter. Since diameter of is and the number of vertices in is,

6 94 Venkanagouda M. Goudar et al hence by theorem., 1. We have , Theorem 3.13 For any path, 3, 3 4, 3. Proof. Let be formed from two copies of of. Then by Theorem.3. Now formed from two copies of, of.and let,,,,,,. We have the following cases. Case1. If, then by the definition of line graph,. By theorem.3. Case. If 3, then is formed from two copies of clearly 3. Case3. Suppose 3. Let S be the geodetic set of. We claim that S contains two elements (end vertices) from each set,,,. Since, it follows that 4. It remains to show that if is a three element subset of then. First assume that is a subset U or W, say the farmer. Then. Therefore, we may take that,. Then,. Theorem 3.14 For the wheel, 6,n is even,. Proof. Let 6 with x the vertex of and,,,,, be the internal edges of. Now,,, are the vertices formed from edges of. i.e.,,, are the vertices of formed from internal edges of. i.e.,. Now forms a minimum geodetic set of ), Clearly.. Theorem 3.15 For the wheel 6, n is odd,.

7 On the geodetic number of line graph 95 Proof. Let 6 with x the vertex of and,,,,, be the internal edges of. Now,,, are the vertices formed from edges of. i.e.,,, are the vertices of formed from internal edges of. i.e.,. Now, forms a minimum geodetic set of ), clearly,. References [1] F Buckley and F. Harary Distance in graphs, Addison-Wesely, Reading, MA (1990). [] G. Chartrand, F. Harary, and P.Zhang. Geodetic sets in graphs Discussiones Mathematicae Graph Theory 0 (000), [3] G. Chartrand, F. Harary, H.C Swart and P.Zhang, Geodomination in graphs, Bull. ICA 31(001), [4] G. Chartrand, F. Harary, and P.Zhang, on the geodetic number of a graph.networks.39(00) 1-6. [5] G. Chartrand and P.Zhang, Introduction to Graph Theory, Tata McGraw Hill Pub.Co.Ltd.(006). [6] F.Harary, Graph Theory, Addison-Wesely, Reading, MA,(1969). Received: September, 01