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arXiv · 2005.08095

General $d$-position sets

Sandi Klavzar·Douglas F. Rall·Ismael G. Yero
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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.

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Sandi Klavzar, Douglas F. Rall, Ismael G. Yero. 2020-05-16. General $d$-position sets. https://arxiv.org/abs/2005.08095

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