Greedy defining sets of graphs

For a graph G and an order a on V(G), we define a greedy defining set as a subset S of V(G) with an assignment of colors to vertices in S, such that the pre-coloring can be extended to a x( G)-coloring of G by the greedy coloring of (G, a). A greedy defining set ofaX( G)-coloring C of G is a greedy defining set, which results in the coloring C (by the greedy procedure). We denote the size of a greedy defining set of C with minimum cardinality by G D N (G, a, C). In this paper we show that the problem of determining GDN(G,a,C), for an instance (G,a,C) is an NP-complete problem.