We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixed-parameter intractable. 1 Hypertree Decompositions: Definition and Basics This paper reports about the recently introduced concept of hypertr...

Abstract This chapter introduces three hypergraph computation paradigms, including intra-hypergraph computation, inter-hypergraph and structure computation. Intra-hypergraph representation aims to conduct learning of a hypergraph, where each subject is represented by its components. Inter-hypergraph vertices in the vertex hypergraph. Hypergraph prediction, which find connections among vertices....

Tree-width of graphs is a well studied notion, which plays an important role in structural graph theory and has many algorithmic applications. Various other graph invariants are known to be the same or within a constant factor of tree-width, for example, the bramble number or tangle number of a graph [4, 5], the branch-width [5], the linkedness [4], and the number of cops required to win the ro...

In this paper, the concept of hypergraph is extended to the fuzzy hypergraph. In the fuzzy hypergraph, the concepts of a-cut hypergraph, strength of edge and dual fuzzy hypergraph are developed. It is shown that the fuzzy hypergraph and a-cut hypergraph are useful to represent a fuzzy partition. An application example also shows that the strength of edge can be used to decompose the data set in...

Abstract Hypergraph modeling is the fundamental task in hypergraph computation, which targets on establishing a high-quality structure to accurately formulate high-order correlation among data. In this section, we introduce different methods show how build hypergraphs using various pieces of information, such as features, attributes, and/or graphs. These are organized into two broad categories,...

A k-uniform hypergraphH = (V ;E) is said to be self-complementary whenever it is isomorphic with its complement H = (V ; ` V k ́ −E). Every permutation σ of the set V such that σ(e) is an edge ofH if and only if e ∈ E is called selfcomplementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel (1963) and Sachs (1962). For any positive inte...

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph pLaplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to ana...

With the advent of big data and information age, magnitude various complex networks is growing rapidly. Many real-life situations cannot be portrayed by ordinary networks, while hypergraphs have ability to describe characterize higher order relationships, which attracted extensive attention from academia industry in recent years. Firstly, this paper described development process, application ar...