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For a graph $G$ and an order $\sigma$ 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 ${\cal X}(G)$-coloring of $G$ by the greedy coloring
of $(G,\sigma)$. A greedy defining set of a ${\cal X}(G)$-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 $GDN(G,\sigma,C)$.
In this paper we show that the problem of determining
$GDN(G,\sigma,C)$, for an instance $(G,\sigma,C)$ is an
$NP$-complete problem.
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