Bounds on the connected local dimension of graphs in terms of the marked dimension and the clique number

Abstract

Let G be a connected graph and let v be a vertex of G. The representation of v with respect to an ordered set is the k-vector where is a distance between v and wi for If the representations of any two adjacent vertices of G with respect to W are distinct and the induced subgraph is connected, then W is called a connected local resolving set of G. The minimum cardinality of connected local resolving sets of G is referred to as the connected local dimension of G, denoted by A connected local resolving set of cardinality is called a minimum connected local resolving set or a connected local basis of G. The true twin graph tG of G is obtained by true twin equivalence classes of G such that the vertex set of tG consists of every true twin equivalence class of G and any two distinct vertices of tG are adjacent if the distance of them in G is 1. A connected local resolving set of tG containing all marked vertices is called a marked set of tG. A marked set of tG having minimum cardinality is called a minimum marked set or a marked basis of tG and this cardinality is called the marked dimension of tG, which is denoted by In this work, we investigate the connected local dimension of G by using the marked dimension of its true twin graph tG. The bounds for the connected local dimension of G are presented in terms of the marked dimension of tG and the clique number of a set of all marked vertices of tG.