On the edge geodetic and k-edge geodetic number of a graph

For vertices u and v in a connected graph G = (V, E), the distance d(u, v) is the length of a shortest u - v path in G. A u - v path of length d(u, v) is called a u - v geodesic. For an integer ..., a geodesic of length k in G is called a k-geodesic. A set ... is a k-edge geodetic set of G if each edge ... lies on a k-geodesic of some pair of vertices in S and a set T Ç V is an edge geodetic set of G if each edge of G lies on a geodesic of some pair of vertices in T, and Smarandache edge-geodetic set of G if each edge of G lies on at least two geodesics of T. The minimum cardinality of a k-edge geodetic set of G is the k-edge geodetic number egk(G) and the minimum cardinality of an edge geodetic set is the edge geodetic number eg(G). In this paper we investigate how the edge geodetic number and the k-edge geodetic number of a graph G are affected by adding a pendant edge to G. It is proved that if G' is a graph obtained from G by adding a pendant edge, then ... and ... For any integer ..., it is also proved that ... It is shown that for any integer ... and for every pair a, b of integers with ..., there is a connected graph G such that egk(G) = b and egk(G′) = a, where G′ is a graph obtained from G by adding a pendant edge.