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Let  G  be a connected graph and let  v  be a vertex of  G . The representation of  v  with respect to an ordered setis the  k -vectorwhereis a distance between  v  and  w iforIf the representations of any two adjacent vertices of  G  with respect to  W  are distinct and the induced subgraphis 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 byA connected local resolving set of cardinalityis 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 byIn 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 .

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