Truncated metric dimension for finite graphs

Publication Date

10-30-2022

Document Type

Article

Publication Title

Discrete Applied Mathematics

Volume

320

DOI

10.1016/j.dam.2022.04.021

First Page

150

Last Page

169

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.

Funding Number

1836914

Funding Sponsor

National Science Foundation

Keywords

Adjacency dimension, Distance-k dimension, k-truncated metric dimension, k-truncated resolving set, Metric dimension, Truncated metric dimension

Department

Mathematics and Statistics

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