  Graph theoretical problems based on shortest paths are at the core of
research due to their theoretical importance and applicability. This paper
deals with the geodetic number which is a global measure for simple connected
graphs and it belongs to the path covering problems: what is the
minimal-cardinality set of vertices, such that all shortest paths between its
elements cover every vertex of the graph. Inspired by the exact 0-1 integer
linear programming formalism from the recent literature, we propose a new
methods to obtain upper bounds for the geodetic number in an algorithmic way.
The efficiency of these algorithms are demonstrated on a collection of
structurally different graphs.
