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),Aguilóet 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.
