Article in volume
Authors:
Sandi Klavžar
Faculty of Mathematics and PhysicsUniversity of LjubljanaJadranska 191000 LjubljanaSlovenia
email: sandi.klavzar@fmf.uni-lj.si
Aditi Krishnakumar
School of Mathematics and Statistics, Open University, Milton Keynes, UK
email: aditikrishnakumar@gmail.com
James Tuite
School of Mathematics and Statistics, Open University, Milton Keynes, UK
email: james.t.tuite@open.ac.uk
Title:
Lower general position sets in graphs
PDFSource:
Discussiones Mathematicae Graph Theory 45(2) (2025) 509-531
Received: 2023-06-16 , Revised: 2024-02-13 , Accepted: 2024-02-15 , Available online: 2024-03-11 , https://doi.org/10.7151/dmgt.2542
Abstract:
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 gp$^-(G)$ of $G$, which is the number of
vertices in a smallest maximal general position set of $G$. We show that
$\textrm{gp}^-(G) = 2$ 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.
Keywords:
general position number, geodetic number, universal line, computational complexity, Kneser graphs, line graphs
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