Short note of SuperHyperClique-width and Local Superhypertree-width

Abstract

Tree-width is a fundamental parameter that quantifies how ”tree-like” an undirected graph is,
 based on its optimal tree decomposition [32]. Several related concepts, including Hypertree-width [20], Branch
width [30], Linear-width [5], Local Tree-width [22], and SuperHypertree-width [13], have been extensively
 studied. Clique-width, on the other hand, is defined as the minimum number of labels required to construct a
 graph using four operations: vertex creation, disjoint union, edge insertion, and relabeling [24]. Local tree-width
 is a function mapping radius r to the maximum tree-width among all r-neighborhood induced subgraphs in a
 graph. A hypergraph is a generalization of a graph where each edge can connect any number of vertices, not
 just two. The concept of a SuperHyperGraph generalizes the classical notion of a hypergraph by introducing
 recursive hierarchical structures.
 In this paper, we introduce new graph parameters: HyperClique-width, SuperHyperClique-width, Local
 Hypertree-width, and Local SuperHypertree-width. We formally define these parameters and provide an initial
 mathematical exploration of their structural properties.

DOI: 10.5281/zenodo.15540424