On the Vertex Position Number of Graphs

TL;DR

This paper introduces the concept of vertex position sets in graphs, analyzing their sizes across different graph classes, and explores the computational complexity of identifying such sets.

Contribution

It generalizes the notion of visibility to graph theory and provides bounds and exact values for maximum vertex position sets in various graph classes.

Findings

Determined maximum and minimum sizes of vertex position sets for common graph classes.

Established bounds based on girth, degrees, diameter, and radius.

Discussed the complexity of finding maximum vertex position sets.

Abstract

In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.