  Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any
subset of $X$. Also denote the power set of $X$ by $\mathcal{P}(X)$. An integer
additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to
\mathcal{P}(X)$ such that the induced function $f^+:E(G) \to \mathcal{P}(X)$ is
defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$
and $f(v)$. An IASL $f$ is said to be a topological IASL (Top-IASL) if
$f(V(G))\cup \{\emptyset\}$ is a topology of the ground set $X$. An IASL is
said to be an integer additive set-graceful labeling (IASGL) if for the induced
edge-function $f^+$, $f^+(E(G))= \mathcal{P}(X)-\{\emptyset, \{0\}\}$. In this
paper, we study certain types of IASL of a given graph $G$, which is a
topological integer additive set-labeling as well as an integer additive
set-graceful labeling of $G$.
