In this paper, we develop a divide-and-conquer approach, called block decomposition, to
solve the minimum geodetic set problem. This provides us with a unified approach for all
graphs admitting blocks for which the problem of finding a minimum geodetic set containing
a given set of vertices (g-extension problem) can be efficiently solved. Our
method allows us to derive linear time algorithms for the minimum geodetic set problem in
(a proper superclass of) block-cacti and monopolar chordal graphs. Also, we show that hull
sets and geodetic sets of block-cacti are the same, and the minimum geodetic set problem
is NP-hard in cobipartite graphs. We conclude by pointing out several interesting research
directions.