Critical Sets for Sudoku and General Graphs

TL;DR

This paper explores the concept of critical sets in graphs, especially in Sudoku graphs, analyzing their properties, computational complexity, and bounds, and introduces new parameters related to extremal critical sets.

Contribution

It defines four new parameters for critical sets, proves their properties, establishes computational intractability, and computes exact values for certain graph classes.

Findings

Critical sets are computationally intractable to find.

Exact values of parameters are computed for specific graph classes.

Bounds are established for generalized Sudoku graphs.

Abstract

We discuss the problem of finding critical sets in graphs, a concept which has appeared in a number of guises in the combinatorics and graph theory literature. The case of the Sudoku graph receives particular attention, because critical sets correspond to minimal fair puzzles. We define four parameters associated with the sizes of extremal critical sets and (a) prove several general results about these parameters' properties, including their computational intractability, (b) compute their values exactly for some classes of graphs, (c) obtain bounds for generalized Sudoku graphs, and (d) offer a number of open questions regarding critical sets and the aforementioned parameters.