Characterizing Bipartite Graphs which Admit k-NU Polymorphisms via Absolute Retracts

TL;DR

This paper characterizes bipartite graphs that admit a (k+1)-ary near-unanimity polymorphism by introducing bipartite absolute retracts related to tree obstructions, extending previous results for specific cases.

Contribution

It establishes a precise correspondence between bipartite absolute retracts with respect to tree obstructions and graphs admitting (k+1)-NU polymorphisms, generalizing earlier findings.

Findings

Bipartite absolute retracts are characterized via tree obstructions with at most k leaves.

Bipartite graphs with a (k+1)-NU polymorphism are exactly the bipartite absolute retracts.

The result generalizes known cases for 3-NU polymorphisms and reflexive graphs.

Abstract

We first introduce the class of bipartite absolute retracts with respect to tree obstructions with at most $k$ leaves. Then, using the theory of homomorphism duality, we show that this class of absolute retracts coincides exactly with the bipartite graphs which admit a $(k+1)$-ary near-unanimity (NU) polymorphism. This result mirrors the case for reflexive graphs and generalizes a known result for bipartite graphs admitting a $3$-NU polymorphism.