Leírás
We consider intersection hypergraphs defined on a finite family S of n (possibly intersecting) axis-parallel segments by a finite family of pairwise disjoint curves C such that every curve has an endpoint which lies in the same connected region of \(\mathbb{R}^2\setminus S\). We call such a family of curves grounded. We show that such a hypergraph has at most \(O(n)\) hyperedges of size 2. We further show, using a general framework, how this implies further properties of such a hypergraph: it has \(O(k^c n)\) hyperedges of size at most \(k\) and can be properly colored with \(O(1)\) colors. These results imply respective results about intersection hypergraphs defined on a finite family of grounded L-shapes by another finite family of grounded L-shapes (an L-shape is grounded if its top endpoint lies on the \(x\)-axis), improving a recent result of Keller, Rok and Smorodinsky.
Joint work with Eyal Ackerman and Dömötör Pálvölgyi.
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