Weak Set-Labeling Number of Certain IASL-Graphs

Let ${\mathbb{N}}_0$ be the set of all non-negative integers, let $X\subset $\mathbb{N}$_0$and$$\mathcal{P}$(X)$bethethepowersetof$X$. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to $\mathcal{P}$($\mathbb{N}$_0)$suchthattheinducedfunction$f^+:E(G) \to $\mathcal{P}$($\mathbb{N}$_0)$isdefinedby$f^+ (uv) = f(u)+ f(v)$,where$f(u)+f(v)$isthesumsetof$f(u)$and$f(v)$\mathrm{.}AnIASL$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 $\mathrm{\mid }{f}^{+}(uv)\mathrm{\mid }=\max (\mathrm{\mid }f(u)\mathrm{\mid },\mathrm{\mid }f(v)\mathrm{\mid })\mathrm{\forall }uv\in E(G)$. 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.