On General Position Sets in Cartesian Products

The general position number gp ( G ) of a connected graph G is the cardinality of a largest set S of vertices such that no three distinct vertices from S lie on a common geodesic; such sets are refereed to as gp-sets of G . The general position number of cylinders P r □ C s is deduced. It is proved...

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Bibliographic Details
Published in:Resultate der Mathematik Vol. 76; no. 3
Main Authors: Klavžar, Sandi, Patkós, Balázs, Rus, Gregor, Yero, Ismael G.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.08.2021
Subjects:
Mathematics
Mathematics and Statistics
05C76
probabilistic constructions
05C12
05C30
05D40
General position problem
paths and cycles
Cartesian product of graphs
exact enumeration
ISSN:1422-6383, 1420-9012
Online Access:Get full text
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Summary:The general position number gp ( G ) of a connected graph G is the cardinality of a largest set S of vertices such that no three distinct vertices from S lie on a common geodesic; such sets are refereed to as gp-sets of G . The general position number of cylinders P r □ C s is deduced. It is proved that gp ( C r □ C s ) ∈ { 6 , 7 } whenever r ≥ s ≥ 3 , s ≠ 4 , and r ≥ 6 . A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in P r □ P s , where r , s ≥ 2 , is also determined.
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-021-01438-x