Graphs with the edge metric dimension smaller than the metric dimension

作者：

Highlights：

• Three problems regarding metric and edge metric dimension of a graph are considered: 1. For which integers r,t,n larger than 0 with r,t being at most n-1, can be constructed a graph G of order n with dim(G)=r and edim(G)=t? 2. Is it possible that dim(G) or edim(G) would be bounded from above by some constant factor of edim(G) or dim(G), respectively? 3. Can you construct some other families of graphs for which dim(G)>edim(G)?

• We present an almost complete answer to question 1.

• We show the unboundedness in problem 2.

• We give positive answer to question 3.

摘要

•Three problems regarding metric and edge metric dimension of a graph are considered: 1. For which integers r,t,n larger than 0 with r,t being at most n-1, can be constructed a graph G of order n with dim(G)=r and edim(G)=t? 2. Is it possible that dim(G) or edim(G) would be bounded from above by some constant factor of edim(G) or dim(G), respectively? 3. Can you construct some other families of graphs for which dim(G)>edim(G)?•We present an almost complete answer to question 1.•We show the unboundedness in problem 2.•We give positive answer to question 3.

论文关键词：Edge metric dimension,Metric dimension,Unicyclic graphs

论文评审过程：Received 16 June 2020, Revised 2 February 2021, Accepted 7 February 2021, Available online 26 February 2021, Version of Record 26 February 2021.

论文官网地址：https://doi.org/10.1016/j.amc.2021.126076