International J.Math. Combin. Vol.2(2016), 109-120 Nonsplit Geodetic Number of a Graph Tejaswini K M, Venkanagouda M Goudar Sri Gauthama Research Centre, (Affiliated to Kuvempu University, Shimoga) Department of Mathematics, Sri Siddhartha Institute of Technology, Tumkur-572 105, Karnataka, India Venkatesh Department of mathematics, Kuvempu University, Shimoga, India E-mail: tejaswini.ssit@gmail.com, vmgouda@gmail.com vensprema@gmail.com Abstract: Let G be a graph. If u, v ∈ V (G), a u − v geodesic of G is the shortest path between u and v. The closed interval I[u, v] consists of all vertices lying in some u - v geodesic of G. For S ⊆ V (G) the set I[S] is the union of all sets I[u, v] for u, v ∈ S . A set S is a geodetic set of G if I (S)= V (G). The cardinality of a minimum geodetic set of G is the geodetic number of G, denoted by g(G). In this paper, we study the nonsplit geodetic number of a graph gns (G). The set S ⊆ V (G) is a nonsplit geodetic set in G if S is a geodetic set and 〈V (G) − S〉 is connected, nonsplit geodetic number gns (G) of G is the minimum cardinality of a nonsplit geodetic set of G. We investigate the relationship between nonsplit geodetic number and geodetic number. We also obtain the nonsplit geodetic number in the cartesian product of graphs. Key Words: Cartesian products, distance, edge covering number, Smarandachely k- geodetic set, geodetic number, vertex covering number. AMS(2010): 05C05, 05C12. §1. Introduction As usual n = |V | and m = |E| denote the number of vertices and edges of a graph G respectively. The graphs considered here are finite, undirected,simple and connected. 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. It is well known that this distance is a metric on the vertex set V (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 radius, rad G and the maximum eccentricity is the diameter, diam G. A u − v path of length d(u, v) is called a u − v geodesic. We define I[u, v] to the set (interval) of all vertices lying on some u − v geodesic of G and for a nonempty subset S of V (G), I [S ]= ∪ u,v∈S I [u, v]. A set S of vertices of G is called a geodetic set in G if I [S]= V (G), and a geodetic set of minimum cardinality is a minimum geodetic set, and generally, if there is a k-subset T of V (G) such that I (S)  T = V (G), where 0 ≤ k< |G|−|S|, then S is called a Smarandachely k-geodetic set of G. The cardinality of a minimum geodetic set in G is called 1 Received December 3, 2014, Accepted December 6, 2015.