More results on greedy defining sets

TL;DR

This paper investigates the computational complexity of greedy defining sets in graphs, establishes NP-completeness for bipartite graphs, provides polynomial algorithms for forests, and applies these concepts to Latin squares and secret sharing schemes.

Contribution

It proves NP-completeness for bipartite graphs, offers polynomial solutions for forests, and introduces a method for Latin squares with applications in secret sharing.

Findings

NP-complete for bipartite graphs

Polynomial time for forests

Latin squares have GDS of size at most n^2 - (n log n)/4

Abstract

The greedy defining sets of graphs were appeared first time in [M. Zaker, Greedy defining sets of graphs, Australas. J. Combin, 2001]. We show that to determine the greedy defining number of bipartite graphs is an NP-complete problem. This result answers affirmatively the problem mentioned in the previous paper. It is also shown that this number for forests can be determined in polynomial time. Then we present a method for obtaining greedy defining sets in Latin squares and using this method, show that any $n\times n$ Latin square has a GDS of size at most $n^2-(n\log n)/4$. Finally we present an application of greedy defining sets in designing practical secret sharing schemes.