Circular representation problem on hypergraphs
A hypergraph (V,E) is called circular-representable if there exists a bijection from the vertex set into a topological circle such that the vertices of every edge determine an interval with respect to the circular order at the circle (i.e. they are consecutive points on the circle). Necessary and sufficient conditions for the circular representability of a hypergraph are proved. Those circular-representable hypergraphs with a unique cicular representation (up to symmetries of the circle) are characterized. A group of transformations is constructed which acts transitively on the set of all circular representations of a hypergraph; this gives a theoretical overview over all different representations. This is applied to obtain a similar result for interval-representable hypergraphs (which are definded by the corresponding property for the line). Finally relations to the problem of characterizing circular graphs are established: With graphs G of a restricted class of graphs, a hypergraph H(G) is associated in a certain way such that G is circular iff H(G) is circular representable.
scientific article; zbMATH DE number 3873376