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

TL;DR

This paper investigates the general position and mutual-visibility problems in double graphs and Mycielskian graphs, establishing bounds and exact values for these properties in various graph classes.

Contribution

It provides new bounds and exact values for mutual-visibility and general position numbers in double and Mycielskian graphs, expanding understanding of these properties.

Findings

Sharp bounds for mutual-visibility and general position numbers.

Exact mutual-visibility number for double graphs and Mycielskian of cycles.

Relationships between mutual-visibility and total mutual-visibility numbers.

Abstract

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.