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Lower general position sets in graphsGabriele Di Stefano,
Sandi Klavžar,
Aditi Krishnakumar,
James Tuite,
Ismael G. Yero, 2025, izvirni znanstveni članek
Povzetek: 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.
Ključne besede: general position number, geodetic number, universal line, computational complexity, Kneser graphs, line graphs
Objavljeno v DiRROS: 11.04.2025; Ogledov: 152; Prenosov: 68
Celotno besedilo (254,31 KB)
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