On commutative characterization of generalized comb and corona products of graphs with respect to the local metric dimension

Let G be a connected graph with vertex set V(G) and W = {w₁, w₂, ., w_m} ⊂ V(G). The representation of a vertex v ∈ V(G), with respect to W is the ordered m-tuple r(v |W) = (d(v, w₁), d(v, w₂), ., d(v, w_m)), where d(v, w) represents the distance between vertices v and w. This set W is called a local resolving set for G if every two adjacent vertices have a distinct representation and a minimum local resolving set is called a local basis of G. The cardinality of a local basis of G is called the local metric dimension of G, denoted by dim_l (G). In general, comb product and corona product are non-commutative operations in graphs. However, these operations can be made commutative with respect to local metric dimension for some graphs with certain conditions. In this paper, we determine the local metric dimension of generalized comb and corona products of graphs and obtain necessary and sufficient conditions of graphs in order that comb and corona products be commutative operations with respect to the local metric dimension.