Weak Integer Additive Set-Indexed Graphs: A Creative Review

Abstract

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 $f^{\ast }:E(G)\to \mathcal{P}(X)-\{\varnothing \}$ 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.

18 pages, review paper, submitted