Maximal resolving sets in a graph

Abstract

Let G be a connected graph. A subset [Formula: see text] of [Formula: see text] is called a resolving set of G if the code of any vertex [Formula: see text] with respect to S is different from the code of any other vertex where code of u with respect to S denoted by [Formula: see text] is defined as [Formula: see text]. Resolving set was earlier studied in the name of locating set by Slater and Harary and Melter too studied this concept. The minimum cardinality of a resolving set is called the metric dimension (locating number). A vertex [Formula: see text] in a connected graph G is said to resolve two vertices [Formula: see text] if [Formula: see text] Clearly, x resolves [Formula: see text] A subset S of [Formula: see text] is a resolving set of G if for any two distinct vertices [Formula: see text] there exists a vertex [Formula: see text] such that x resolves [Formula: see text] Motivated by this equivalent definition, a study of resolving chain and maximal resolving set is initiated in this paper. Also, study of total resolving sets is initiated.