Tree-Vertex Graph: New Hierarcal Graph Class

A hypergraph generalizes an ordinary graph by allowing an edge to connect any nonempty subset of the vertex set. By iterating the powerset operation one step further, one obtains nested (higherorder) vertex objects and, consequently, a finite SuperHyperGraph whose vertices and edges may themselves be set-valued at multiple levels. Thus, many hierarchical graph structures exist in the literature. Moreover, not only in graph theory but also in broader fields—such as through concepts like Decision Trees and Tree Soft Sets—it is well known that tree structures are effective tools for representing hierarchical concepts. In this paper, we define a new class of graphs called Tree-Vertex Graphs. In this framework, a tree structure is imposed on the vertex set, and the edge set is defined in a manner consistent with the tree-structured vertex set. The tree structure therefore serves as a key concept for representing hierarchical graphs.