Mutual-visibility and general position in double graphs and in Mycielskians

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.