Abstract
Hypergraphs can be partitioned into two classes: tree (acyclic) hypergraphs and cyclic hypergraphs. This paper analyzes a new class of cyclic hypergraphs called Xrings. HypergraphHis an Xring if the hyperedges ofHcan be circularly ordered so that for every vertex, all hyperedges containing the vertex are consecutive; in addition, the vertices of no hyperedge may be a subset of the vertices of another hyperedge, and no vertex may appear in exactly one hyperedge. LetH1andH2be two hypergraphs. A tree projection ofH1with respect toH2is an acyclic hypergraphH3such that the vertices of each hyperedge inH1appear among the vertices of some hyperedge ofH3and the vertices of each hyperedge inH3appear among the vertices of some hyperedge ofH2. A polynomial time algorithm is presented for deciding, given XringH1and arbitrary hypergraphH2, whether there exists a tree projection ofH2with respect toH1. It is shown that hypergraphHis an Xring iff a modified adjacency graph ofHis a circular-arc graph. A linear time Xring recognition algorithm, for GYO reduced hypergraphs as inputs, is presented.
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