The edge geodetic vertex covering number of a graph

For a connected graph G of order n ≥ 2, a set S ⊆V(G) is an edge geodetic vertex cover of G if S is both an edge geodetic set and a vertex covering set of G. The minimum cardinality of an edge geodetic vertex cover of G is defined as the edge geodetic vertex covering number of G and is denoted by g 1α (G). Any edge geodetic vertex cover of cardinality g 1α (G) is a g1α - set of G. Some general properties satisfied by edge geodetic vertex cover are studied. The edge geodetic vertex covering number of several classes of graphs are determined. Connected graphs of order n with edge geodetic vertex covering number 2 is characterized. A few realization results are given for the parameter g 1α (G).