Hypertree

Hypertree

A hypergraph "H" is called a hypertree, if it has a host graph "T" such that "T" is a tree and every hyperedge of "H" induces a subtree in "T". V. I. Voloshin (2002) "Coloring Mixed Hypergraphs", ISBN 0821828126 , [http://books.google.com/books?id=RYM_Qi5HR68C&pg=PA26&dq=hypertree+hypergraph&ei=TX7vSJiuOZHEMaT4pA4&sig=ACfU3U12X-utDYS4BRfgYOOvlCAUCFV-gQ#PPA23,M1 p. 23] ]

Since a tree is a hypertree, hypertrees may be seen as a generalization of the notion of a tree for hypergraphs. Any hypertree is isomorphic to some family of subtrees of a tree.

Properties

A hypertree has the Helly property (2-Helly property), i.e., if any two hyperedges from a subset of its hyperedges have a common vertex, then all hyperedges of the subset have a common vertex.

The line graph of a hypertree is a chordal graph.

A hypergraph is a hypertree if and only if its dual hypergraph is conformal and chordal.

References


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