A Note on the Sparing Number of the Sieve Graphs of Certain Graphs

Let ${\mathbb{N}}_0$ denote the set of all non-negative integers and $\mathcal{P}({\mathbb{N}}_0)$ be its power set. An integer additive set-indexer (IASI) of a given 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)$definedby$f^+ (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ of a graph $G$ is said to be a weak IASI of $G$ if $\mathrm{\mid }{f}^{+}(uv)\mathrm{\mid }=\max (\mathrm{\mid }f(u)\mathrm{\mid },\mathrm{\mid }f(v)\mathrm{\mid })$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels required for a graph $G$ to admit a weak IASI. In this paper, we introduce the notion of $k$-sieve graphs of a given graph and study their sparing numbers.