http://openproblemgarden.org/op/straight_line_representation_of_planar_linear_hypergraphs Comments for "Straight line representation of planar linear hypergraphs" en http://openproblemgarden.org/op/straight_line_representation_of_planar_linear_hypergraphs <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/ossona_de_mendez_p">Ossona de Mendez</a>; <a href="/category/de_fraysseix_h">de Fraysseix</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/graph_theory">Graph Theory</a> » <a href="/category/topological_graph_theory">Topological G.T.</a> » <a href="/category/drawings_0">Drawings</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Every planar linear hypergraph <img class="teximage" src="/files/tex/f5f238182fe95fc868919d496bbfdb7ca814b109.png" alt="$ \mathcal H $" /> has a straight line representation in the plane which maps each vertex <img class="teximage" src="/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png" alt="$ v $" /> to a point <img class="teximage" src="/files/tex/32bfe1e260483a2ab29178080e321fc752a317ea.png" alt="$ p(v) $" /> and each edge <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> to a straight line segment <img class="teximage" src="/files/tex/61bb791a2023293f7ed962e6f97a4c2c1d06c835.png" alt="$ s(E) $" />, in such a way that:<br /> <ul class="itemize"> \item for each vertex <img class="teximage" src="/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png" alt="$ v $" /> and each edge <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" />, we have: <img class="teximage" src="/files/tex/c9f390a742578a9cd85bacba27f9c5c60f3ef8df.png" alt="$$p(v)\in s(E)\quad\iff\quad v\in E$$" /> \item for each couple of distinct edges <img class="teximage" src="/files/tex/b262080a51c9f5cdbcc1cec7f532c72ab0325bee.png" alt="$ E_1,E_2 $" />, we have <img class="teximage" src="/files/tex/a00a1355e84e670f7c5210b6c9adb15850d9ad20.png" alt="$$s(E_1)\cap s(E_2)=\{p(v): v\in E_1\cap E_2\}$$" /> </ul> </div> </tr></td> </table> </td> </tr> </table> de Fraysseix, Hubert Ossona de Mendez, Patrice intersection graph planar hypergraph Graph Theory Topological Graph Theory Drawings http://openproblemgarden.org/op/straight_line_representation_of_planar_linear_hypergraphs#comment Tue, 04 Sep 2007 10:22:16 +0200 taxipom 554 at http://openproblemgarden.org