Structural characterization of some family of local distance pattern distinguishing 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) ρ′ (G). If G admits a LDPD-set, then G is called a LDPD-graph. In this paper, we present the structure characteriation of some family of LDPD-graphs. Also, we discuss the LDPD-sets and LDPD -number in graph products.

Subject Classification: (2010):

• 05C12

Keywords:

• LDPD-set
• LDPD-number