5 research outputs found
Strong geodetic problem on Cartesian products of graphs
The strong geodetic problem is a recent variation of the geodetic problem.
For a graph , its strong geodetic number is the cardinality of
a smallest vertex subset , such that each vertex of lies on a fixed
shortest path between a pair of vertices from . In this paper, the strong
geodetic problem is studied on the Cartesian product of graphs. A general upper
bound for is determined, as well as exact values
for , , and certain prisms.
Connections between the strong geodetic number of a graph and its subgraphs are
also discussed.Comment: 18 pages, 9 figure
Products of Geodesic Graphs and the Geodetic Number of Products
Given a connected graph and a vertex , the geodesic graph has the same vertex set as with edges iff either is on an geodesic path or is on an geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of with itself, where , is equal to the minimum of and eight