Topological Integer Additive Set-Graceful Graphs

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)$suchthattheinducedfunction$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 $f(V(G))\cup \{\mathrm{\varnothing }\}$ 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 $f^{+}$, $f^{+}(E(G))=\mathcal{P}(X)-\{\mathrm{\varnothing },\{0\}\}$. 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 $G$.