Strong Geodetic Number of Complete Bipartite Graphs and of Graphs with Specified Diameter

Abstract

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on one fixed geodesic between a pair of vertices from S. In this paper, some general properties of the strong geodesic problem are studied, especially in connection with diameter of a graph. The problem is also solved for balanced complete bipartite graphs.

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

“…The strong geodetic problem appears to be difficult even on complete bipartite graphs. Some initial investigation is done in [17], where the problem is presented as an optimization problem and the solution is found for balanced complete bipartite graphs. Some more results have been very recently presented in [19], where it is proved that the problem is NP-complete on general bipartite graphs, but polynomial on complete bipartite graphs.…”

“…Notice that m ≥ 4 as n ≥ 2m − 1, m even.On the other hand, 8n + 9 < (m + 1) 2 + 8 < (m + 2) 2 as m ≥ 4m is odd, sg a < sg b . But if m is even, then m + 1 is odd and by[17, Lemma 2.2], there exists an integer k such that (m + 1) 2 = 8k + 1. Clearly, 8n + 9 ≤ (m+1) 2 +7, but due to the congruences modulo 8, we conclude that 8n+9 ≤ (m+1) 2 .…”

Strong geodetic cores and Cartesian product graphs

“…The strong geodetic number was further investigated on grids and cylinders, and on general Cartesian product graphs in [15,18], respectively. Additional properties, in particular with respect to the diameter, and a solution for balanced complete bipartite graphs were reported in [14]. The edge version of the problem has also been introduced and investigated in [21].…”

Strong geodetic problem on complete multipartite graphs

“…In the first paper [16], it was proved that the problem is NP-complete. The invariant has also been determined for complete Apollonian networks [16], thin grids and cylinders [13], and balanced complete bipartite graphs [11]. Some properties of the strong geodetic number of Cartesian product of graphs have been studied in [12].…”

“…Recall from [11] that the strong geodetic problem on a complete bipartite graph can be presented as a (non-linear) optimization problem as follows. Let (X, Y ) be the bipartition of K n 1 ,n 2 and S = S 1 ∪ S 2 , S 1 ⊆ X, S 2 ⊆ Y , its strong geodetic set.…”