Procreating tiles of double commutative-step digraphs

Abstract

Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L(l, h, x, y), Aguiló et al. define a discrete iteration L(p) = L(l + 2p, h + 2p, x + p, y + p), p = 0, 1, 2, …, over L-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by L(l, h, x, y), which is said to be a procreating k-tight tile if L(p)(p = 0, 1, 2, …) are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L(l, h, x, y), 0 ≤ |y − x| ≤ 2k + 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches.