Minimal complete Shidoku symmetry groups

TL;DR

This paper identifies minimal symmetry groups for Shidoku, simplifying the calculation of equivalence classes by reducing the symmetry group without losing classification accuracy, paving the way for similar reductions in larger Sudoku variants.

Contribution

It demonstrates that the standard symmetry group for Shidoku can be minimized to smaller subgroups that preserve the classification of boards, using computational and graph connectivity methods.

Findings

Standard symmetry group can be reduced to minimal subgroups

Minimal groups induce the same equivalence classes as the full group

Method may extend to larger Sudoku variants

Abstract

Calculations of the number of equivalence classes of Sudoku boards has to this point been done only with the aid of a computer, in part because of the unnecessarily large symmetry group used to form the classes. In particular, the relationship between relabeling symmetries and positional symmetries such as row/column swaps is complicated. In this paper we focus first on the smaller Shidoku case and show first by computation and then by using connectivity properties of simple graphs that the usual symmetry group can in fact be reduced to various minimal subgroups that induce the same action. This is the first step in finding a similar reduction in the larger Sudoku case and for other variants of Sudoku.