Timeline for Can Tutte embedding be guaranteed that each face is convex?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2021 at 18:30 | vote | accept | Licheng Zhang | ||
| Apr 13, 2021 at 11:42 | answer | added | Gordon Royle | timeline score: 3 | |
| Apr 13, 2021 at 5:21 | history | edited | Licheng Zhang | CC BY-SA 4.0 |
added 75 characters in body
|
| Apr 13, 2021 at 5:15 | history | edited | Licheng Zhang | CC BY-SA 4.0 |
added 75 characters in body
|
| Apr 13, 2021 at 5:07 | comment | added | Licheng Zhang | @ Brendan McKay Thanks! For 3 connected plane graph, Besides that outside convex face, Is any inner face also convex in Tutte embedding? This is my confusion. | |
| Apr 13, 2021 at 4:27 | comment | added | Brendan McKay | Tutte's method works with any face on the outside drawn as any convex polygon. It doesn't matter which face you choose. But this assumes 3-connectivity. Without 3-connectivity there might not be any Tutte drawing at all. For a simple example, suppose there is a cut-vertex -- then some face has a vertex appearing twice so it is impossible to draw it convex. | |
| Apr 13, 2021 at 4:02 | comment | added | Licheng Zhang | @SamHopkins haa! I personally feel you are right. That is to say, any 3-connected planar graph has a convex staight line drawing which external face is anyone in face set. This is great! Of course, is any Tttue embeddings convex, I don’t know and very curious. | |
| Apr 13, 2021 at 3:50 | comment | added | Sam Hopkins | Shouldn't the Schlegel diagram of the polyhedron achieve this? | |
| Apr 13, 2021 at 3:46 | comment | added | Licheng Zhang | @SamHopkins Steinitz's theorem seems to only say that there is a correspondence between a 3-connected plane graph and a convex polyhedron, but I am not sure that given an any convex(may be not) external face, it can guarantee that there is a plane straight-line drawing where anyinternal face is all convex. Maybe I don't understand it well. | |
| Apr 13, 2021 at 3:36 | comment | added | Licheng Zhang | @GordonRoyle Thank you very much for your answers. I already understand what you mean, for this example, there really is no such drawing in Tutte embemdding. This is just an example of mine. Maybe not good. I don’t know if any non-convex face can also be excluded for the general situation. | |
| Apr 13, 2021 at 3:32 | comment | added | Sam Hopkins | You might also be interested in Steinitz's theorem. | |
| Apr 13, 2021 at 3:30 | comment | added | Gordon Royle | $w$ will be placed midway along the line connecting $e$ and $z$, so the face will never appear as you have drawn it. | |
| Apr 13, 2021 at 3:24 | history | asked | Licheng Zhang | CC BY-SA 4.0 |