The general position problem and strong resolving graphs

Abstract

The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp(G) ≥ ω(G SR , where G SR is the strong resolving graph of G, and ω(G SR ) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs.

On General Position Sets in Cartesian Products

“…Following the seminal papers, the general position problem has been investigated in a sequence of papers [1,6,10,14,16,18,22,26]. As it happens, in the special case of hypercubes, the general position problem was studied back in 1995 by Körner [11] related to some coding theory problems.…”

Zero forcing number versus general position number in tree-like graphs

“…In [3], general position sets in graphs were characterized. Several additional papers on the concept followed, many of them dealing with bounds on the general position number and exact results in product graphs, Kneser graphs, and more, see [11,17,18,22,24,[26][27][28]. In addition, the concept was very recently extended to the Steiner general position number [16].…”

Further new results on strong resolving partitions for graphs

“…For more information on the strong resolving graph of a graph (as a proper graph operation), we suggest a previous study [23], where several structural properties of such graphs were studied, and the recent work [24] which gives some new results concerning strong resolving graphs.…”