Truncated metric dimension for finite graphs

Abstract

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