Abstract
A vertex subset S of a graph is called a strong geodetic set if there exists a choice of exactly one geodesic for each pair of vertices of S in such a way that these (|S| 2) geodesics cover all the vertices of graph G. The strong geodetic number of G, denoted by sg(G), is the smallest cardinality of a strong geodetic set. In this paper, we give an upper bound of strong geodetic number of the Cartesian product graphs and study this parameter for some Cartesian product networks.
Highlights
All graphs considered in this paper are connected, simple, undirected and finite
We give an upper bound of strong geodetic number of the Cartesian product graphs and study this parameter for some Cartesian product networks
The strong geodetic problem is to find a minimum strong geodetic set S of G, the minimum cardinality of a strong geodetic set is defined as the strong geodetic number, denoted by sg(G)
Summary
All graphs considered in this paper are connected, simple, undirected and finite. We refer to the book (Bondy & Murty, 2008) for graph theoretical notation and terminology not described here. The strong geodetic problem is to find a minimum strong geodetic set S of G, the minimum cardinality of a strong geodetic set is defined as the strong geodetic number, denoted by sg(G) Later, this concept and other related invariants in various classes of graph are considered in several literature (Atici, 2002; Bresar et al, 2011; Bresar & Tepeh, 2008; Chartrand et al, 2002; Fitzpatrick, 1999; Manuel et al, 2017; Ye et al, 2007). The Cartesian product of two graphs G and H, denoted by G2H, is a graph with vertex set V(G) × V(H) such that (u, v) and (u′, v′) are adjacent if and only if either u = u′ and vv′ ∈ E(H), or v = v′ and uu′ ∈ E(G). We study the exact value of sg(Qn) for n ≤ 5 and an upper bound when n ≥ 6
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