Strong Geodetic Number of Complete Bipartite Graphs, Crown Graphs and Hypercubes

Abstract

The strong geodetic number, sg(G), of a graph G is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for sg(K n,m ) is given, as well as a formula for the crown graphs S 0 n . Bounds on sg(Q n ) are also discussed.

An O(mn2) algorithm for computing the strong geodetic number in outerplanar graphs

“…In the example above, we have already seen that X = {0, 6, 8} is an MGS of the graph in Figure 1. It is immediate to check that the paths induced by Arc[0, 6], Arc [6,8] and Arc [8,0] are geodesics. So X is also an MSGS of G.…”

“…In order to model a problem on social networks, a variant to the GNP has been introduced in [2] (see also [9,[11][12][13] and [8] where the state of the art on the strong geodetic number is summarized). Given a set S of supervisors of a social network, we suppose that these supervisors can communicate each other by using a single, fixed shortest path through the network.…”

Strong edge geodetic problem on grids

“…The exact value of the strong geodetic number was computed for different families of graphs. For example, for complete bipartite graphs K n,m it was first computed for cases when n = m, and for n ≫ m in [4], and later in the general case in [1]. In [6], the strong geodetic number was computed for some balanced multipartite complete graphs and it was shown that computing the strong geodetic number of general complete multipartite graphs is NP-complete.…”

“…In [6], the strong geodetic number was computed for some balanced multipartite complete graphs and it was shown that computing the strong geodetic number of general complete multipartite graphs is NP-complete. The exact strong geodetic number was also computed for crown graphs S n 0 in [1], Hamming graphs K m K n in [5], Cartesian products K 1,n P l in [5], thin (n ≫ m) grids P n P m , and thin (n ≫ m) cylinders P n C m in [7], and i-level complete Appolonian networks A(i) in [9].…”

“…[2] proved an upper bound on the strong geodetic number of Cartesian product graphs using the so-called strong geodetic core number. In [1], an upper bound on the strong edge geodetic number was given for hypercubes. A general upper bound was in [5] computed for the strong geodetic number of Cartesian product graphs and an upper bound for the strong geodetic number of the Cartesian product of a path with an arbitrary graph was computed in [7].…”