Unbounded fault-tolerant local metric dimension at local dimension two

Construct or rule out a family of graphs G with local metric dimension ldim(G)=2 for which the fault-tolerant local metric dimension ftldim(G) is unbounded.

Background

The paper characterizes graphs with ldim(G) equal to 0 or 1 and correspondingly characterizes ftldim(G) equal to 1 or 2. These results imply a finite bound on fault-tolerant local metric dimension when local metric dimension is at most 1. The authors explain that existing results do not determine whether such a bound persists at local metric dimension 2.