For a non-empty ground set $X$, finite or infinite, the { \em set-valuation} or { \em set-labeling} of a given graph $G$ is an injective function $f:V(G) \to \mathcal{ P}(X)$, where $\mathcal{ P}(X)$ is the power set of the set $X$. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to \mathcal{ P}(X)$ such that the function $f^{ \ast}:E(G)\to \mathcal{ P}(X)-\{ \emptyset\}$ defined by $f^{ \ast}(uv) = f(u){ \ast} f(v)$ for every $uv{ \in} E(G)$ is also injective., where $\ast$ is a binary operation on sets. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{ P}({ \mathbb{ N}_0})$ such that the induced function $g_f:E(G) \to \mathcal{ P}(\mathbb{ N}_0)$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{ N}_0$ is the set of all non-negative integers and $\mathcal{ P}(\mathbb{ N}_0)$ is its power set. A weak IASI is an IASI $f$ such that $|f^+(uv)|= \text{ max}(f(u),f(v))$. In this paper, we critically and creatively review the concepts and properties of weak integer additive set-valued graphs.
Categories: math.CO