A review on graceful and sequential integer additive set-labeled graphs

Let $ \mathbb N _0 $ be the set of all non-negative integers and $ \fancyscript {P} $ be its power set. Then, an integer additive set-labeling (IASL) of a graph G is an injective function $ f:V(G)\rightarrow \fancyscript {P}( \mathbb N _0) $, such that the induced function $ f^+:E(G) \rightarrow \fancyscript {P}( \mathbb N _0) $ defined by $ f^+(uv)=f(u)+f(v) $ and an integer additive set-indexer (IASI) is an integer additive set-labeling such that the induced function $ f^+:E(G) \rightarrow \fancyscript {P}( \mathbb N _0) $ is also injective. An integer additive set-labeling (or an integer additive set-indexer) is said to be a graceful integer additive set-labeling (or graceful integer additive set-indexer) if $ f^{+}(E(G))= \fancyscript {P}(X)-\{\emptyset ,\{0\}\} $ and an integer additive set-labeling (or an integer additive set-indexer) is said to be a sequential integer additive set-labeling (or sequential integer additive set-indexer) if $ f(V(G))\cup f^{+}(E(G))= \fancyscript {P}(X)-\{\emptyset \} $. In this write-up, we creatively and critically review certain studies made on graceful integer additive set-labeling and sequential integer additive set-labeling of given graphs and certain properties and characteristics of the graphs which satisfy this type of set-labelings