Let G = (V, E) be a connected graph and W ? V be a nonempty set. For each u ∈ V, the set f_W (u) = {d(u, v) ∶ v ∈ W} is called the distance pattern of u with respect to the set W. If f_W (x) ≠ f_W (y) for all xy ∈ E(G), then W is called a local distance pattern distinguishing set (or a LDPD-set in short) of G. The minimum cardinality of a LDPD-set in G, if it exists, is the LDPD-number of G and is denoted by ρ′(G). If G admits a LDPD-set, then G is called a LDPD-graph. In this paper we discuss the LDPD-number ρ′(G) of some family of graphs and the relation between ρ′(G) and other graph theoretic parameters. We characterized several family of graphs which admits LDPD-sets.
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