Simple Polyominoes are Prime
Ayesha Asloob Qureshi, Takafumi Shibuta, Akihiro Shikama

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.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
