Constructions of 1-Uniform dcsl graphs using Well-graded families of sets

A 1-uniform dcsl of a graph G   is an injective set assignment function f:V(G)→2Xf:V(G)→2X, X be a non-empty set, such that the corresponding induced function f⊕:V(G)×V(G)→2X\{ϕ}f⊕:V(G)×V(G)→2X\{ϕ} given by f⊕(uv)=f(u)⊕f(v)f⊕(uv)=f(u)⊕f(v) satisfies |f⊕(u,v)|=1.d(u,v)|f⊕(u,v)|=1.d(u,v) for all distinct u,v∈V(G)u,v∈V(G), where d(u,v)d(u,v) is the length of a shortest path between u and v  , and f(u)⊕f(v)f(u)⊕f(v) denotes the symmetric difference of the two sets. Let FF be a family of subsets of a set X. A tight path between two distinct sets P and Q (or from P to Q  ) in FF is a sequence P0=P,P1,P2…Pn=QP0=P,P1,P2…Pn=Q in FF such that d(P,Q)=|PΔQ|=nd(P,Q)=|PΔQ|=n and d(Pi,Pi+1)=1d(Pi,Pi+1)=1 for 0≤i≤n−10≤i≤n−1. The family FF is well-graded (or wg-family), if there is a tight path   between any two of its distinct sets. Any family FF of subsets of X   defines a graph GF=(F,EF)GF=(F,EF), where EF={{P,Q}⊆F:|PΔQ|=1}EF={{P,Q}⊆F:|PΔQ|=1}, and we call GFGF, an  FF-induced graph  . The purpose of this paper is to examine the existence of 1-uniform dcsl of an induced graph GF1∪F2∪…FnGF1∪F2∪…Fn formed from the finite union of well-graded families F1,F2,…F1,F2,…, and FnFn by introducing amalgamation techniques in between them, where, for 1≤i≤n1≤i≤n, each FiFi-induced graph, GFiGFi is isomorphic to a 1-uniform dcsl even cycle.