Summary: Hypertree Width and Related Hypergraph Invariants
Isolde Adler Georg Gottlob Martin Grohe
5th May 2006
We study the notion of hypertree width of hypergraphs. We prove that, up to a constant factor,
hypertree width is the same as a number of other hypergraph invariants that resemble graph invariants
such as bramble number, branch width, linkedness, and the minimum number of cops required to win
Seymour and Thomas's robber and cops game.
Tree width of graphs is a well studied notion, which plays an important role in structural graph theory and
has many algorithmic applications. Various other graph invariants are known to be the same or within a
constant factor of tree width, for example, the bramble number or tangle number of a graph [12, 13], the
branch width , the linkedness , and the number of cops required to win the robber and cops game
on the graph . Several of these notions may be viewed as measures for the global connectivity of a
graph. The various equivalent characterizations of tree width show that it is a natural and robust notion.
Formally, let us call two graph or hypergraph invariants I and J equivalent if they are within a constant
factor of each other, that is, if there are constants c,d > 0 such that for all graphs or hypergraphs G we have
c·I(G) J(G) d ·I(G).
Tree decompositions and tree width can be generalized to hypergraphs in a straightforward manner; the
tree width of a hypergraph is equal to the tree width of its primal graph. Motivated by algorithmic problems