Applied Mathematics and Scientific Computing pp 517-525 | Cite as

# 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.

## Keywords

LDPD-set LDPD-number Local metric set## Notes

### Acknowledgements

The author is thankful to Professor S. Arumugam for suggesting this problem. Also the author is very much grateful to the constant support of MEPCO Schlenk Engineering College (Autonomous), Sivakasi.

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