Families of Graceful Spiders with 3ℓ, 3ℓ + 2 and 3ℓ – 1 Legs

We say that a tree is a spider if has at most one vertex of degree greater than two. We prove the existence of families of graceful spiders with 3ℓ, 3ℓ + 2 and 3ℓ − 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.