The upper geodetic vertex covering number of a graph

A set S ⊆ V (G) is a geodetic vertex cover of G if S is both a geodetic set and a vertex cover of G. The minimum cardinality of a geodetic vertex cover of G is defined as the geodetic vertex covering number of G and is denoted by gα(G) . A geodetic vertex cover S in a connected graph G is called a minimal geodetic vertex cover of G if no proper subset of S is a geodetic vertex cover of G. The upper geodetic vertex covering number g+­­α(G) of G is the maximum cardinality of a minimal geodetic vertex cover of G. Some general properties satisfied by the upper geodetic vertex covering number of a graph are studied. The upper geodetic vertex covering number of several classes of graphs are determined. Some bounds for g+­­α(G) are obtained and the graphs attaining these bounds are characterized.