![](/files/happy5.png)
Ossona de Mendez, Patrice
Linear Hypergraphs with Dimension 3 ★★
Author(s): de Fraysseix; Ossona de Mendez; Rosenstiehl
Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.
Keywords: Hypergraphs
Straight line representation of planar linear hypergraphs ★★
Author(s): de Fraysseix; Ossona de Mendez
Conjecture Every planar linear hypergraph
has a straight line representation in the plane which maps each vertex
to a point
and each edge
to a straight line segment
, in such a way that:
![$ \mathcal H $](/files/tex/f5f238182fe95fc868919d496bbfdb7ca814b109.png)
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
![$ p(v) $](/files/tex/32bfe1e260483a2ab29178080e321fc752a317ea.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ s(E) $](/files/tex/61bb791a2023293f7ed962e6f97a4c2c1d06c835.png)
- \item for each vertex
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$$p(v)\in s(E)\quad\iff\quad v\in E$$](/files/tex/c9f390a742578a9cd85bacba27f9c5c60f3ef8df.png)
![$ E_1,E_2 $](/files/tex/b262080a51c9f5cdbcc1cec7f532c72ab0325bee.png)
![$$s(E_1)\cap s(E_2)=\{p(v): v\in E_1\cap E_2\}$$](/files/tex/a00a1355e84e670f7c5210b6c9adb15850d9ad20.png)
Keywords: intersection graph; planar hypergraph
![Syndicate content Syndicate content](/misc/feed.png)