Characterization of general position sets and its applications to cographs and bipartite graphs

Abstract

A vertex subset S of a graph G is a general position set of G if no vertex of S lies on a geodesic between two other vertices of S. The cardinality of a largest general position set of G is the general position number gp(G) of G. It is proved that S ⊆ V (G) is in general position if and only if the components of G[S] are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of S. If diam(G) = 2, then gp(G) is the maximum of ω(G) and the maximum order of an induced complete multipartite subgraph of the complement of G. As a consequence, gp(G) of a cograph G can be determined in polynomial time. If G is bipartite, then gp(G) ≤ α(G) with equality if diam(G) ∈ {2, 3}. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.1

On General Position Sets in Cartesian Products

“…Concerning gp (G) we wonder whether one can write limit instead of limit superior in the definition of gp (G). Moreover, by the above we have lim k→∞ p(C k ) = 1 4 . We also pose: the copyright holder.…”

“…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.…”

On independent position sets in graphs

“…But the same concept has already been studied two years earlier in [5] under the name geodetic irredundant sets. Refer [6,7,8,9,10,11] to understand the recent developments on general position number.…”

The general position problem and strong resolving graphs

“…If G = (V (G), E(G)) is a graph, then S ⊆ V (G) is a general position set if no triple of vertices from S lie on a common geodesic in G. The general position problem is to find a largest general position set of G, the order of such a set is the general position number gp(G) of G. A general position set of G of order gp(G) is shortly called gp-set. The general position problem has been further studied in a sequence of very recent papers [1,2,8,10].…”