General position and mutual-visibility in shadow graphs

TL;DR

This paper studies the maximum size of vertex sets in shadow graphs that satisfy conditions related to shortest paths, providing bounds, characterizations, and exact values for various graph classes.

Contribution

It establishes sharp bounds and characterizations for the general position and mutual-visibility numbers in shadow graphs, including for standard graph classes.

Findings

Sharp bounds for general position and mutual-visibility numbers.

Characterizations of extremal shadow graphs.

Exact values for cycles, multipartite graphs, and trees.

Abstract

The \emph{general position problem} in graphs asks for a largest set of vertices in which no three lie on a common shortest path. The \emph{mutual-visibility problem} seeks a largest set of vertices such that every pair is connected by a shortest path whose internal vertices lie outside the set. In this paper, we investigate the general position and mutual-visibility problems for shadow graphs. Sharp general bounds are established for both the general position number and the mutual-visibility number of shadow graphs, and classes of graphs attaining these extremal values are characterized. Furthermore, these invariants are determined for several standard classes of shadow graphs, including shadow graphs of cycles, multipartite graphs, and trees.