  Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any
non-empty subset of $\mathbb{N}_0$. Denote the power set of $X$ by
$\mathcal{P}(X)$. An integer additive set-labeling (IASL) of a graph $G$ is an
injective set-valued 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 $f(v)$. If the associated
set-valued edge function $f^+$ is also injective, then such an IASL is called
an integer additive set-indexer (IASI). An IASL $f$ is said to be a topological
IASL (TIASL) if $f(V(G))\cup \{\emptyset\}$ is a topology of the ground set
$X$. An IASL is said to be an integer additive set-sequential labeling (IASSL)
if $f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}$. An IASL of a given
graph $G$ is said to be a topological integer additive set-sequential labeling
of $G$, if it is a topological integer additive set-labeling as well as an
integer additive set-sequential labeling of $G$. In this paper, we study the
conditions required for a graph $G$ to admit this type of IASL and propose some
important characteristics of the graphs which admit this type of IASLs.
