Lower general position sets in graphs

Di Stefano, Gabriele (author), Klavžar, Sandi (author), Krishnakumar, Aditi (author), Tuite, James (author), Yero, Ismael G. (author)

A subset $S$ of vertices of a graph $G$ is a general position set if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the lower general position number ${\rm gp}^-(G)$ of $G$, which is the number of vertices in a smallest maximal general position set of $G$. We show that ${\rm gp}^-(G) = 2$ if and only if $G$ contains a universal line and determine this number for several classes of graphs, including Kneser graphs $K(n,2)$, line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.

• Language: English
• Keywords: general position number, geodetic number, universal line, computational complexity, Kneser graphs, line graphs
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2025
• Year: 2025
• Number of pages: Str. 509-531
• Numbering: Vol. 45, no. 2
• PID: 20.500.12556/RUL-168403
• UDC: 519.17
• ISSN on article: 1234-3099
• DOI: 10.7151/dmgt.2542
• COBISS.SI-ID: 232294915
• Publication date in RUL: 11.04.2025

Record is a part of a journal

• Title: Discussiones mathematicae : Graph theory
• Shortened title: Discuss. Math., Graph Theory
• Publisher: Technical University Press
• ISSN: 1234-3099
• COBISS.SI-ID: 7487065

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: število splošne lege, geodetsko število, univerzalna premica, računska zahtevnost, Kneserjevi grafi, grafi povezav