General d-position sets

Abstract

The general d-position number gp(d)(G) of a graph G is the cardinality of a largest set S for which no three distinct vertices from S lie on a common geodesic of length at most d. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of gp(d)(G) with respect to the suitable values of d. We show that the decision problem concerning finding gp(d)(G) is NP-complete for any value of d. The value of gp(d)(G) when G is a path or a cycle is computed and a structural characterization of general d-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that gp(d)(G) is infinite whenever G is an infinite graph and d is a finite integer.