The 2-edge geodetic number and graph operations

For a connected graph G = ( V , E ) of order n ≥ 2, a set is a 2- edge geodetic set of G if each edge lies on a u - v geodesic with d ( u , v ) = 2 for some vertices u and v in S . The minimum cardinality of a 2-edge geodetic set in G is the 2- edge geodetic number of G , denoted by eg 2 ( G ). It is proved that for any connected graph G , β 1 ( G ) ≤ eg 2 ( G ), where β 1 ( G ) is the matching number of G . It is shown that every pair a , b of integers with 2 ≤ a ≤ b is realizable as the matching number and 2-edge geodetic number, respectively, of some connected graph. We determine bounds for the 2-edge geodetic number of Cartesian product of graphs. Also we determine the 2-edge geodetic number of certain classes of Cartesian product graphs. The 2-edge geodetic number of join of two graphs is obtained in terms of the 2-edge geodetic number of the factor graphs.