  Let $\mathbb{N}_0$ be the set of all non-negative integers, let $X\subset
\mathbb{N}_0$ and $\mathcal{P}(X)$ be the the power set of $X$. An integer
additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to
\mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to
\mathcal{P}(\mathbb{N}_0)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where
$f(u)+f(v)$ is the sum set of $f(u)$ and $f(v)$. An IASL $f$ is said to be an
integer additive set-indexer (IASI) of a graph $G$ if the induced edge function
$f^+$ is also injective. An integer additive set-labeling $f$ is said to be a
weak integer additive set-labeling (WIASL) if
$|f^+(uv)|=\max(|f(u)|,|f(v)|)~\forall ~ uv\in E(G)$. The minimum cardinality
of the ground set $X$ required for a given graph $G$ to admit an IASL is called
the set-labeling number of the graph. In this paper, we introduce the notion of
the weak set-labeling number of a graph $G$ as the minimum cardinality of $X$
so that $G$ admits a WIASL with respect to the ground set $X$ and discuss the
weak set-labeling number of certain graphs.
