Communities in graphs and hypergraphs

Abstract

In this paper we define a type of cohesive subgroups -called communities -in hypergraphs, based on the edge connectivity of subhypergraphs. We describe a simple algorithm for the construction of these sets and show, based on examples from image segmentation and information retrieval, that these groups may be useful for the analysis and accessibility of large graphs and hypergraphs.

Subdivision Drawings of Hypergraphs

“…Hence, modulo re-labeling of vertices and the removal of white non-vertex faces, we must have a subdivision drawing as the one depicted in Fig. 5(a) with a non-simple hyperedge region for hyperedge (2,3,4). Second, consider the hypergraph H 2 that is schematically depicted in Fig.…”

“…In relational databases there is a natural correspondence between database schemata and hypergraphs, with attributes corresponding to vertices and relations to hyperedges [7]. Hypergraphs are used in VLSI design for circuit visualization [6,14] and also appear in computational biology [10,12] and social networks [3].…”

“…2(a, b)). Concerning vertex-planarity, consider the following hypergraph H: H has six vertices and three hyperedges (1, 2, 3, 4), (1,2,3,5), and (1, 2, 3, 6). H is vertex-planar but not (Zykov-)planar.…”

On Planar Supports for Hypergraphs

“…For example, there is a natural correspondence between hypergraphs and database schemata in relational databases, with vertices corresponding to attributes and hyperedges to relations (e.g., see [2]). Further applications include VLSI design [13], computational biology [12], and social networks [5].…”

“…For example, vertex-based Venn diagrams [9] and concrete Euler diagrams [7] are both subdivision drawings. (2,3,4,5), (1,3,4,6,7), (6,7,8,9)}.…”