20.500.12556/DiRROS-20785 Variety of general position problems in graphs 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. general position total general position outer general position dual general position Cartesian product of graphs strong resolving graph convex subgraph splošna lega celotna splošna lega zunanja splošna lega dualna splošna lega kartezični produkt grafov krepki solventni graf konveksen podgraf true false true Angleški jezik Slovenski jezik Neznano 2024-11-07 12:21:13 2024-11-07 12:21:13 2024-11-20 08:19:54 0000-00-00 00:00:00 2025 0 0 14 str. iss. 1, [article no.] 5 Vol. 48 Jan. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 0126-6705 10.1007/s40840-024-01788-z 213851395 s40840-024-01788-z-1.pdf s40840-024-01788-z-1.pdf 1 6B93719702121058EF4162871595D6E9 d615463cf9ed00514fc5fe053c1355016d97fb6bd91c8cd1f00a740ee6078094 adcf66d4-9cfa-11ef-89f3-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=29339 https://link.springer.com/article/10.1007/s40840-024-01788-z 1 6726a951-9cfa-11ef-89f3-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=29338 Inštitut za matematiko, fiziko in mehaniko 0 0 0