Changes

== Abstract ==

== Abstract ==

In a given graph $G$, a set of vertices $S$ with an assignment of

In a given graph $G$, a set of vertices $S$ with an assignment of

colors is called a {\em defining set (of a $k$--coloring)}, if there

colors is called a ''defining set'' (of a $k$--coloring), if there

exists a unique extension of the colors of $S$ to a proper $k$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\em minimum

exists a unique extension of the colors of $S$ to a proper $k$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a ''minimum

defining set.} The cardinality of minimum defining set is the

defining set''. The cardinality of minimum defining set is the

{\em defining number} denoted by {$d(G, k)$}. A {\em critical set} is a minimal defining set.  

''defining number'' denoted by {$d(G, k)$}. A ''critical set'' is a minimal defining set.  

Defining sets are defined and discussed for many concepts and parameters in graph theory and combinatorics.

Defining sets are defined and discussed for many concepts and parameters in graph theory and combinatorics.

For example in Latin squares a {\em{critical set}} is a partial Latin square that has a unique

For example in Latin squares a ''critical set'' is a partial Latin square that has a unique

completion to a Latin square, and is minimal with respect to this

completion to a Latin square, and is minimal with respect to this

property. Smallest possible size of a critical set in any Latin

property. Smallest possible size of a critical set in any Latin