D-interval hypergraph

Not to be confused with interval graph.

In graph theory, a d-interval hypergraph is a kind of a hypergraph constructed using intervals of real lines. The parameter d is a positive integer. The vertices of a d-interval hypergraph are the points of d disjoint lines (thus there are uncountably many vertices). The edges of the graph are d-tuples of intervals, one interval in every real line.[1]

The simplest case is d = 1. The vertex set of a 1-interval hypergraph is the set of real numbers; each edge in such a hypergraph is an interval of the real line. For example, the set { [−2, −1], [0, 5], [3, 7] } defines a 1-interval hypergraph. Note the difference from an interval graph: in an interval graph, the vertices are the intervals (a finite set); in a 1-interval hypergraph, the vertices are all points in the real line (an uncountable set).

As another example, in a 2-interval hypergraph, the vertex set is the disjoint union of two real lines, and each edge is a union of two intervals: one in line #1 and one in line #2.

The following two concepts are defined for d-interval hypergraphs just like for finite hypergraphs:

  • A matching is a set of non-intersecting edges, i.e., a set of non-intersecting d-intervals. For example, in the 1-interval hypergraph { [−2, −1], [0, 5], [3, 7] }, the set { [−2, −1], [0, 5] } is a matching of size 2, but the set { [0, 5], [3, 7] } is not a matching since its elements intersect. The largest matching size in H is denoted by ν(H).
  • A covering or a transversal is a set of vertices that intersects every edge, i.e., a set of points that intersects every d-interval. For example, in the 1-interval hypergraph { [−2, −1], [0, 5], [3, 7] }, the set {−1.5, 4} is a covering of size 2, but the set {−1.5, 1} is not a covering since it does not intersect the edge [3, 7]. The smallest transversal size in H is denoted by τ(H).

ν(H) ≤ τ(H) is true for any hypergraph H.

Tibor Gallai proved that, in a 1-interval hypergraph, they are equal: τ(H) = ν(H). This is analogous to Kőnig's theorem for bipartite graphs.

Gabor Tardos[1] proved that, in a 2-interval hypergraph, τ(H) ≤ 2ν(H), and it is tight (i.e., every 2-interval hypergraph with a matching of size m, can be covered by 2m points).

Kaiser[2] proved that, in a d-interval hypergraph, τ(H) ≤ d(d – 1)ν(H), and moreover, every d-interval hypergraph with a matching of size m, can be covered by at d(d − 1)m points, (d − 1)m points on each line.

Frick and Zerbib[3] proved a colorful ("rainbow") version of this theorem.

  1. ^ a b Tardos, Gábor (1995-03-01). "Transversals of 2-intervals, a topological approach". Combinatorica. 15 (1): 123–134. doi:10.1007/bf01294464. ISSN 0209-9683.
  2. ^ Kaiser, T. (1997-09-01). "Transversals of d-Intervals". Discrete & Computational Geometry. 18 (2): 195–203. doi:10.1007/PL00009315. ISSN 1432-0444.
  3. ^ Frick, Florian; Zerbib, Shira (2019-06-01). "Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals". Combinatorica. 39 (3): 627–637. arXiv:1710.07722. doi:10.1007/s00493-018-3891-1. ISSN 1439-6912.