On the Properties and Illustrative Examples of Soft SuperHypergraphs
and Rough SuperHypergraphs
Takaaki Fujita1,∗ , Atiqe Ur Rahman2, Arkan A. Ghaib3, Talal Ali Al-Hawary4,
Arif Mehmood Khattak5
1Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan
2Department of Mathematics, University of Management and Technology, Lahore, 54000, Pakistan
3Department of Information Technology, Management Technical College, Southern Technical University,
Basrah, 61004, Iraq
4Department of Mathematics, Yarmouk University, Irbid, Jordan
5Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050,
KPK, Pakistan
Emails: Takaaki.fujita060@gmail.com; aurkhb@gmail.com; arkan.ghaib@stu.edu.iq;
talalhawary@yahoo.com; mehdaniyal@gmail.com
Abstract
In graph theory, a hypergraph generalizes a classical graph by allowing each hyperedge to join any num-
ber of vertices, thereby modeling relationships beyond simple pairwise connections.1 A superhypergraph
takes this further by applying recursive powerset constructions to its hyperedge set, creating hierarchical and
self-referential network layers.2 A soft graph defines a family of subgraphs parameterized over a fixed uni-
verse of vertices and edges, while a rough graph uses lower and upper approximations to capture uncertainty
in graph structure. In this paper, we revisit Soft SuperHypergraphs and Rough SuperHypergraphs—originally
introduced in3—which integrate the flexibility of soft and rough graph frameworks with the layered com-
plexity of superhypergraphs. We provide precise definitions, illustrative examples, and a detailed analysis of
their fundamental properties, demonstrating their potential for modeling hierarchical and uncertain network
systems.
Keywords: Superhypergraph; Hypergraph; Soft Graph; Rough Graph; Soft HyperGraph; Rough HyperGraph