Strong Geodetic Number in Some Networks

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.

The Geodetic Number for the Unit Graphs Associated with Rings of Order P and P2

“…Furthermore, Chartrand et al [11] showed that, for any connected graph G, g(G) ≤ n − diam(G) + 1, where dim(G) = max{d(u, v) : u, v ∈ V}. Several researchers have studied this topic and presented some results [12][13][14][15][16][17][18].…”

Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity

“…, n { }as i ∈ [n]. Some of the variants for covering the vertices problem are referred in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Some results on strong geodetic problems can be referred in [15][16][17][18][19].…”

“…Some of the variants for covering the vertices problem are referred in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Some results on strong geodetic problems can be referred in [15][16][17][18][19]. A set S⊆V(G) is a geodetic set if it covers all the vertices of V(G)\S, and the cardinality of smallest geodetic set is called the geodetic number, denoted by g (G).…”