Critical Sets in Edge-Magic Total Labelings

Let \( \lambda \) be an edge-magic total (EMT) labeling of graph \( G(V, E) \). Let \( W \subset V(G) \cup E(G) \). Any restriction of \( \lambda \) to \( W \) is called a \emph{partial EMT labeling} on \( G \). A partial EMT labeling \( \pi \) is a critical set in \( \lambda \) if \( \lambda \) is the only edge-magic total labeling having \( \pi \) as its partial EMT labeling, and no proper restriction of \( \pi \) satisfies the first condition. In this paper, we study the property of critical sets in such a labeling. We determine critical sets in an EMT labeling for a given graph \( G \).