Variety of general position problems in graphs

Tian, Jing (author), Klavžar, Sandi (author)

Let $X$ be a vertex subset of a graph $G$. Then $u, v\in V(G)$ are $X$-positionable if $V(P)\cap X \subseteq \{u,v\}$ holds for any shortest $u,v$-path $P$. If each two vertices from $X$ are $X$-positionable, then $X$ is a general position set. The general position number of $G$ is the cardinality of a largest general position set of $G$ and has been already well investigated. In this paper a variety of general position problems is introduced based on which natural pairs of vertices are required to be $X$-positionable. This yields the total (resp. dual, outer) general position number. It is proved that the total general position sets coincide with sets of simplicial vertices, and that the outer general position sets coincide with sets of mutually maximally distant vertices. It is shown that a general position set is a dual general position set if and only if its complement is convex. Several sufficient conditions are presented that guarantee that a given graph has no dual general position set. The total general position number, the outer general position number, and the dual general position number of arbitrary Cartesian products are determined.

• Language: English
• Keywords: general position, total general position, outer general position, dual general position, Cartesian product of graphs, strong resolving graph, convex subgraph
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 14 str.
• Numbering: Vol. 48, iss. 1, art. 5
• PID: 20.500.12556/RUL-164685
• UDC: 519.17
• ISSN on article: 0126-6705
• DOI: 10.1007/s40840-024-01788-z
• COBISS.SI-ID: 213851395
• Publication date in RUL: 07.11.2024

Record is a part of a journal

• Title: Bulletin of the Malaysian Mathematical Sciences Society
• Shortened title: Bull. Malays. Math. Sci. Soc.
• Publisher: Springer Nature, Malaysian Mathematical Sciences Society, Penerbit Universiti Sains Malaysia
• ISSN: 0126-6705
• COBISS.SI-ID: 515781657

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: splošna lega, celotna splošna lega, zunanja splošna lega, dualna splošna lega, kartezični produkt grafov, krepki solventni graf, konveksni podgraf