<p>In both classical graph theory and fuzzy graph theory, the number of nodes and arcs is conventionally considered fixed. As a result, removing one or more nodes from a graph usually results in a structure that cannot maintain its original function or may no longer be analyzed as the intended graph. To address this limitation, in this paper, we introduce, for the first time, a novel structure called the permanent graph (permanent fuzzy graph), utilizing the established concepts of neighborhood and distance between two nodes. By introducing this particular and useful graph structure, we propose a method to revive a graph whose functionality has been compromised due to the loss of nodes, thereby preventing its complete collapse. The application of this permanent graph (permanent fuzzy graph) in decision-making processes (specifically for making appropriate and informed choices within a social network) serves as evidence supporting this claim.</p>