We introduce the notion of a of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both and are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by X(H). An algorithm for c...

Four consequence operators based on hypergraph satisfiability are defined. Their properties are explored and interconnections are displayed. Finally their relation to the case of the Classical Propositional Calculus is shown. 1. Preliminaries. Let us recall some definitions and facts which can also be found in Cowen [1,2] and Kolany [3]. A hypergraph is a structure G = (V, E), where V is a set ...

Basic definitions. For X a finite set of vertices, a hypergraph H on X is a family of subsets (called edges) of X. In a directed hypergraph H, every set h E H contains a distinguished element called the head of h (a set h may appear more than once in H but with different heads). A hypergraph H’ is a subhypergraph of H iff H’ c H. A directed hypergraph H is an arborescence iff (i) H is empty, or...

ÐIn this work, we show that the standard graph-partitioning-based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrix-vector multiplication. We propose two computational hypergraph models which avoid this crucial deficiency of the graph model. The proposed models reduce the decomposition problem to the well-known hypergraph partition...

We consider a generalization of the notion of transversal to a finite hypergraph, so called weighted transversals. Given a non-negative weight vector assigned to each hyperedge of the input hypergraph, we define a weighted transversal as a minimal vertex set which intersects a collection of hyperedges of sufficiently large total weight. We show that the hypergraph of all weighted transversals i...

This chapter explores original techniques for the construction of hypergraph models for computer vision applications. A hypergraph is a generalization of a pairwise simple graph, where an edge can connect any number of vertices. The expressive power of the hypergraph models places a special emphasis on the relationship among three or more objects, which has made hypergraphs better models of cho...

The cyclicity of a hypergraph is an efficiently computable integer that extends the notion of the cyclomatic number of a graph. Generalizing the notion of the degree of a node in a graph, we define the star articulation degree of a subedge in a hypergraph, and then use it to set up the expression for the cyclicity. The basic properties of cyclicity are that it is zero on acyclic hypergraphs and...