Simple Polyominoes are Prime

TL;DR

This paper establishes a connection between simple polyominoes and weakly chordal bipartite graphs, demonstrating that their associated ideals have quadratic Gr"obner bases, which advances understanding in algebraic combinatorics.

Contribution

It proves that the polyomino ideal of a simple polyomino matches the toric ideal of a weakly chordal bipartite graph, revealing new algebraic properties.

Findings

Polyomino ideal coincides with toric ideal of a weakly chordal bipartite graph

Existence of quadratic Gr"obner basis for these ideals

Enhanced understanding of algebraic structures of simple polyominoes

Abstract

In this paper we show that polyomino ideal of a simple polyomino coincides with the toric ideal of a weakly chordal bipartite graph and hence it has a quadratic Gr\"obner basis with respect to a suitable monomial order.