On the Computational Complexity of Defining Sets

Suppose we have a family ${\cal F}$ of sets. For every $S \in {\cal F}$, a set $D \subseteq S$ is a {\sf defining set} for $({\cal F},S)$ if $S$ is the only element of $\cal{F}$ that contains $D$ as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is $\Sigma_2$-complete. We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is $\Sigma_2$-complete: {\sc Instance:} A graph $G$ with a vertex coloring $c$ and an integer $k$. {\sc Question:} If ${\cal C}(G)$ be the set of all $\chi(G)$-colorings of $G$, then does $({\cal C}(G),c)$ have a defining set of size at most $k$? Moreover, we study the computational complexity of some other variants of this problem

The Size of the Smallest Strong Critical Set in a Latin Square

A critical set in a latin square is a set of entries in a latin square<E-372> which can be embedded in only one latin square. Also, if any element of the<E-384> critical set is deleted, the remaining set can be embedded in more than one latin<E-378> square. A critical set is strong if the embedding latin square is particularly easy<E-379> to find because the remaining squares of the latin square are "forced" one at a<E-392> time. A semi-strong critical set is a generalization of a strong critical set. It is<E-389> proved that the size of the smallest strong or semi-strong critical set of a latin<E-378> square of order n is ën<E-155> 2<E-165> /4û . An example of a critical set that is not strong or<E-388> semi-strong is also displayed. It is also proved that the smallest critical set of a<E-392> latin square of order 6 is 9.<E-172> 1. Introduction<E-89> This paper deals with critical sets in latin squares. A latin square, L, of<E-388> order n is a n x n array with elements chosen fro..