Families of Gracefuls Spiders with ℓ(2k + 1) − k, ℓ(2k + 1) − k + 1 and ℓ(2k + 1) + k + 1 Legs

We say that a tree is a spider if has at most one vertex of degree greater than two. We obtain existence of families of gracefuls spiders with ℓ(2k + 1) − k, ℓ(2k + 1) − k + 1 and ℓ(2k + 1) + k + 1 legs. We provide specific labels for each spider graph, these labels are constructed from graceful path graphs that have a particular label, so there is a correspondence between some paths and graceful spiders that we are studying, this correspondence is described in an algorithm outlined in the preliminaries.