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Variety of general position problems in graphs
ID Tian, Jing (Author), ID Klavžar, Sandi (Author)

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Abstract
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 This link opens in a new window
UDC:519.17
ISSN on article:0126-6705
DOI:10.1007/s40840-024-01788-z This link opens in a new window
COBISS.SI-ID:213851395 This link opens in a new window
Publication date in RUL:07.11.2024
Views:102
Downloads:8
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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 This link opens in a new window

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

Projects

Funder:ARIS - Slovenian Research and Innovation Agency
Project number:P1-0297
Name:Teorija grafov

Funder:ARIS - Slovenian Research and Innovation Agency
Project number:N1-0355
Name:Prirejanja, transverzale in hipergrafi

Funder:ARIS - Slovenian Research and Innovation Agency
Project number:N1-0285
Name:Metrični problemi v grafih in hipergrafih

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