Tree-Related Widths of Graphs and Hypergraphs

A hypergraph pair is a pair (G,H) where G and H are hypergraphs on the same set of vertices. We extend the definitions of hypertree-width [7] and generalised hypertree-width [8] from hypergraphs to hypergraph pairs. We show that for constant k the problem of deciding whether a hypergraph pair has generalised hypertree-width ≤ k, is equivalent to the Hypergraph Sandwich Problem (HSP) [13]. It was recently proved in [9] that the HSP is NP-complete. For constant k there is a polynomial time algorithm that decides whether a given hypergraph pair has hypertree-width ≤ k. (For hypertree-width of hypergraphs, this was shown in [7].) It follows that the HSP is solvable in polynomial time for of inputs (G,H) satisfying: ghw(G,H) ≤ 1 if, and only if, hw(G,H) ≤ 1. Besides this practical interest, hypergraph pairs serve as a tool for giving a common proof for the game theoretic characterisations of tree-width [14] and hypertree-width [8]. Furthermore, they enable us to show a compactness property of generalised hypertree-width for a large class of hypergraphs, the hypergraphs with finite character. Finally, we present two examples showing that neither hypertree-width of hypergraph pairs nor hypertree-width of hypergraphs has the compactness property.