On the Computational Complexity of Defining Sets

TL;DR

This paper investigates the concept of defining sets across various combinatorial structures and proves that determining minimal defining sets is computationally complex, specifically $\,\Sigma_2$-complete, for logical formulas and graph colorings.

Contribution

It introduces the notion of defining sets for logical formulas and establishes their $\,\Sigma_2$-completeness, extending the concept to graph colorings and analyzing related computational complexities.

Findings

Defining sets for logical formulas are $\,\Sigma_2$-complete to compute.

Determining small defining sets for graph colorings is $\,\Sigma_2$-complete.

The study extends the concept of defining sets to new combinatorial contexts.

Abstract

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.