All metric bases and fault-tolerant metric dimension for square of grid

Summary: For a simple connected graph G = ( V, E ) and an ordered subset W = { w 1 , w 2 , . . . , w k } of V , the code of a vertex v ∈ V , denoted by code( v ) , with respect to W is a k -tuple ( d ( v, w 1 ) , . . . , d ( v, w k )) , where d ( v, w t ) represents the distance between v and w t . The set W is called a resolving set of G if code( u ) ̸ = code( v ) for every pair of distinct vertices u and v . A metric basis of G is a resolving set with the minimum cardinality. The metric dimension of G is the cardinality of a metric basis and is denoted by β ( G ) . A set F ⊂ V is called fault-tolerant resolving set of G if F \ { v } is a resolving set of G for every v ∈ F . The fault-tolerant metric dimension of G is the cardinality of a minimal fault-tolerant resolving set. In this article, a complete characterization of metric bases for G 2 mn has been given. In addition, we prove that the fault-tolerant metric dimension of G 2 mn is 4 if m + n is even. We also show that the fault-tolerant metric dimension of G 2 mn is at least 5 and at most 6 when m + n is