Snakes and Caterpillars in Graceful Graphs

Graceful labelings use a prominent place among difference vertex labelings. In this work we present new families of graceful graphs all of them obtained applying a general substitution result. This substitution is applied here to replace some paths with some trees with a more complex structures. Two caterpillars with the same size are said to be textit{analogous} if thelarger stable sets, in both caterpillars, have the same cardinality. We studythe conditions that allow us to replace, within a gracefully labeled graph,some snakes (or paths) by analogous caterpillars, to produce a new gracefulgraph. We present five families of graphs where this replacement isfeasible, generalizing in this way some existing results subdivided trees, first attachment trees, path-like trees, two-point union of paths, and armed crowns.