  The general position problem in graphs is to find the maximum number of
vertices that can be selected such that no three vertices lie on a common
shortest path. The mutual-visibility problem in graphs is to find the maximum
number of vertices that can be selected such that every pair of vertices in the
collection has a shortest path between them with no vertex from the collection
as an internal vertex. In this paper, the general position problem and the
mutual-visibility problem is investigated in double graphs and in Mycielskian
graphs. Sharp general bounds are proved, in particular involving the total
mutual-visibility number and the outer mutual-visibility number of base graphs.
Several exact values are also determined, in particular the mutual-visibility
number of the double graphs and of the Mycielskian of cycles.
