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

The general position problem in graphs is to find the largest possible set of vertices with the property that no three of them 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. Here, the general position problem and the mutual-visibility problem are investigated in double graphs and in Mycielskian graphs. Sharp general bounds are proved, in particular involving the total 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. • The mutual-visibility number of the double of a graph is bounded below using the total mutual-visibility number. • The general position number of the double of a graph is bounded with the general position number of the graph. • The mutual-visibility number of the Mycielskian over a path and over a cycle is determined. • Bounds for the mutual-visibility number of the Mycielskian over a graph of diameter at most three are proved.