A tolled walk ▫$T$▫ between two non-adjacent vertices ▫$u$▫ and ▫$v$▫ in a graph ▫$G$▫ is a walk, in which ▫$u$▫ is adjacent only to the second vertex of ▫$T$▫ and ▫$v$▫ is adjacent only to the ...second-to-last vertex of ▫$T$▫. A toll interval between ▫$u, v \in V(G)$▫ is a set ▫$T_G(u, v) = \{x \in V(G) \vert x$▫ lies on a tolled walk between ▫$u$▫ and ▫$v$▫. A set ▫$S \subseteq V(G)$▫ is toll convex, if ▫$T_G(u, v) \subseteq S$▫ for all ▫$u, v \in S$▫. A toll closure of a set ▫$S \subseteq V(G)$▫ is the union of toll intervals between all pairs of vertices from ▫$S$▫. The size of a smallest set ▫$S$▫ whose toll closure is the whole vertex set is called the toll number of ▫$G$▫, ▫$\operatorname{tn}(G)$▫. This paper investigates the toll number of the strong product of graphs. First, a description of toll intervals between two vertices in the strong product graphs is given. Using this result we characterize graphs with ▫$\operatorname{tn}(G \boxtimes H) = 2$▫ and graphs with ▫$\operatorname{tn}(G \boxtimes H) = 3$▫, which are the only two possibilities. As an addition, for the t-hull number of ▫$G \boxtimes H$▫ we show that ▫$\operatorname{th}(G \boxtimes H) = 2$▫ for any non-complete graphs ▫$G$▫ and ▫$H$▫. As extreme vertices play an important role in different convexity types, we show that no vertex of the strong product graph of two non-complete graphs is an extreme vertex with respect to the toll convexity.
It was not necessary to add the material to the shelf.
Bibliographic material was successfully added on shelf.
Adding bibliographical material on shelf was not successful.
Material is already on the shelf.
Permalink
URL:
Impact factor
Access to the JCR database is permitted only to users from Slovenia. Your current IP address is not on the list of IP addresses with access permission, and authentication with the relevant AAI accout is required.
Year
Impact factor
Edition
Category
Classification
JCR
SNIP
JCR
SNIP
JCR
SNIP
JCR
SNIP
Select the library membership card:
DRS,
in which the journal is indexed
Database name
Field
Year
Links to authors' personal bibliographies
Links to information on researchers in the SICRIS system
Subject headings in COBISS General List of Subject Headings
Select pickup location
The material from the parent unit is free. If the material is delivered to the pickup location from another unit, the library may charge you for this service.
PDF
