Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

TL;DR

This paper develops a canonization procedure in Choiceless Polynomial Time for graphs with bounded dihedral color classes, capturing PTime for this class, including non-abelian cases, by normalizing structures and solving canonization in CPT.

Contribution

It introduces a novel canonization method for graphs with dihedral color classes of bounded size in CPT, capturing PTime on this class, including non-abelian groups, which was previously unresolved.

Findings

Canonization in CPT for graphs with dihedral color classes of bounded size.

First CPT result for non-abelian color classes.

Normal form reduction preserves canonization in CPT.

Abstract

In the quest for a logic capturing PTime the next natural classes of structures to consider are those with bounded color class size. We present a canonization procedure for graphs with dihedral color classes of bounded size in the logic of Choiceless Polynomial Time (CPT), which then captures PTime on this class of structures. This is the first result of this form for non-abelian color classes. The first step proposes a normal form which comprises a "rigid assemblage". This roughly means that the local automorphism groups form 2-injective 3-factor subdirect products. Structures with color classes of bounded size can be reduced canonization preservingly to normal form in CPT. In the second step, we show that for graphs in normal form with dihedral color classes of bounded size, the canonization problem can be solved in CPT. We also show the same statement for general ternary structures in normal form if the dihedral groups are defined over odd domains.