On resolvability of a graph associated to a finite vector space

Abstract

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789-806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set V of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with V as the ground set is also solved.