General $d$-position sets

TL;DR

This paper introduces the general $d$-position number, explores its properties, computational complexity, exact values for paths and cycles, structural characterizations, and its behavior on infinite graphs.

Contribution

It generalizes the concept of position numbers in graphs, analyzes its properties, NP-completeness, and provides exact and structural results for specific graph classes.

Findings

${ m gp}_d(G)$ is NP-complete to compute for any $d$.

Exact values of ${ m gp}_d(G)$ for paths and cycles are determined.

${ m gp}_d(G)$ is infinite for infinite graphs with finite $d$.

Abstract

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural characterization of general $d$-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is infinite whenever $G$ is an infinite graph and $d$ is a finite integer.