Primality of multiply connected polyominoes

Abstract

It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.

Primality of closed path polyominoes

“…At first we show that having an L-configuration or a ladder with at least three steps is a necessary and sufficient condition in order to have no zig-zag walks for a closed path. This result characterizes the structure of closed paths which contain no zig-zag walks, and makes possible to prove that the conjecture in [12] is true for such a class. At the end of this work we study some particular classes of prime polyominoes, that we can build using paths and simple polyominoes.…”

“…Similarly we define vertical edge intervals and maximal vertical edge intervals. A proper interval [a, b] is called an inner interval if all cells of [a, b] belong to P. Following [12], we call zig-zag walk a sequence W : I 1 , . .…”

“…It is a not easy challenge to give a complete classification of all polyominoes whose ideal is prime. In [12] the authors give an interesting tool, which seems to be very useful to characterize the primality of multiply connected polyominoes. They define a particular sequence of inner intervals of P, called a zig-zag walk, and they prove that its no existence in P is a necessary condition to the primality of I P .…”

“…We give some sufficient conditions for the primality of the related ideals. Necessary conditions for primality, and so proving the conjecture of [12] for such classes, are difficult to find out and leave here as open questions.…”

Regularity and Gorenstein property of the $L$-convex Polyominoes

“…It has been shown in [13] and [21] that this is the case if the polyomino is simply connected. In a more recent paper by Mascia, Rinaldo and Romeo [17], it is shown that if K [P] is a domain then it should have no zig-zag walks, and they conjecture that this is also a sufficient condition for K [P] to be a domain. They verified this conjecture computationally for polyominoes of rank ≤ 14.…”