On certain associated graphs of set-valued signed graphs

Abstract

Let $X$ be a non-empty set and let $\Sigma$ be a signed graph, with corresponding underlying graph $G$ and the signature $\sigma$. An injective function $f: V(\Sigma) \rightarrow \mathscr{P}(X)$ is said to be a set-labeling of $\Sigma$ if $f$ is a set-labeling of the underlying graph $G$ and the signature of $\Sigma$ is defined by $\sigma(u v)=(-1)^{|f(u) \oplus f(v)|}$. A signed graph $\Sigma$ together with a set-labeling $f$ is known as a set-labeled signed graph and is denoted by $\Sigma_f$. In this paper, we discuss the characteristics of certain signed graphs associated with given set-valued signed graphs.