Truncated metric dimension for finite graphs

Let G be a graph with vertex set V(G), and let d(x,y) denote the length of shortest path between nodes x y in G. For positive integer k for distinct x,y∈V(G), dk(x,y)=min{d(x,y),k+1} Rk{x,y}={z∈V(G):dk(x,z)≠dk(y,z)}. A subset S⊆V(G) is k-truncated resolving if |S∩Rk{x,y}|≥1 any pair x,y∈V(G). The metric dimension, dimk(G), minimum cardinality over all sets G, usual dimension recovered when k+1 at least diameter We obtain some general bounds dimension. k≥1, we characterize connected graphs order n dimk(G)=n−2 dimk(G)=n−1. j,k≥1, find maximum possible order, degree, clique number, chromatic number dimk(G)=j. determine dimk(G) cycle or path. also examine effect edge deletion on truncated graphs, study various problems related to trees.