Local Distance Pattern Distinguishing Sets in Graphs

Abstract

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.