Topological Integer Additive Set-Sequential Graphs

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 $f(V(G))\cup \{\mathrm{\varnothing }\}$ is a topology of the ground set $X$. An IASL is said to be an integer additive set-sequential labeling (IASSL) if $f(V(G))\cup f^{+}(E(G))=\mathcal{P}(X)-\{\mathrm{\varnothing }\}$. 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.