Primality of weakly connected collections of cells and weakly closed path polyominoes

TL;DR

This paper investigates the primality of ideals associated with weakly connected cell collections and introduces a new class of non-simple polyominoes called weakly closed paths, linking algebraic and graph-theoretic properties.

Contribution

It characterizes the primality of polyomino ideals for weakly closed paths and connects these ideals to toric ideals of weakly chordal bipartite graphs.

Findings

Ideal generated by inner 2-minors corresponds to toric ideal of a weakly chordal bipartite graph.

Primality of polyomino ideals is characterized for weakly closed paths.

Introduces a new class of non-simple polyominoes with algebraic properties.

Abstract

In this paper we study the primality of weakly connected collections of cells, showing that the ideal generated by inner 2-minors attached to a weakly connected and simple collection of cells is the toric ideal of the edge ring of a weakly chordal bipartite graph. As an application of this result we characterize the primality of the polyomino ideals of weakly closed paths, a new class of non simple polyominoes.