Timeline for Does a notion of convex graph make sense?
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| Nov 4, 2011 at 9:29 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 3, 2011 at 19:06 | answer | added | vc-dim | timeline score: 2 | |
| Nov 3, 2011 at 18:24 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 3, 2011 at 15:01 | answer | added | David Eppstein | timeline score: 3 | |
| Nov 3, 2011 at 11:25 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 3, 2011 at 7:08 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 2, 2011 at 22:22 | comment | added | Valerio Capraro | I wrote down those axioms thinking about the unit ball (or the unit cube) in $\mathbb R^n$ and they seem to be sufficient (at least for the purposes that I have in mind) even in higher dimension. I don't know if there is some dependence on the number of paths between two vertices. At some point I was thinking the same thing, but I have no idea how to rephrase the axioms in such a way. | |
| Nov 2, 2011 at 20:44 | comment | added | Chris Leary | This is an intriguing idea. I was wondering if non-planarity of the graph could be a complication. I could be wrong, but it seems that a lot depends on the number of edges, or, the length, of paths between vertices. If so, would it be of benefit to phrase the axioms or definitions in terms of this? | |
| Nov 2, 2011 at 18:17 | comment | added | Valerio Capraro | I don't need that property. But.. is that not automatic by the first property? Yes, you are right, it's not clear if $\gamma$ has an orientation or not. Well, let's say that that symmetry is not required. | |
| Nov 2, 2011 at 17:28 | comment | added | Goldstern | Do you require that for every path (a,b,...,d) in $\Gamma$, also $(d,..., b,a)$ is in $\Gamma$? | |
| Nov 2, 2011 at 15:23 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 2, 2011 at 7:35 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 2, 2011 at 7:34 | comment | added | Valerio Capraro | Indeed, it's already non-trivial (for me) to find a graph verifying the first three properties. For the notion of contractibility you can see arxiv.org/abs/1111.0268 Sec. 2. | |
| Nov 2, 2011 at 7:15 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 2, 2011 at 7:14 | comment | added | Valerio Capraro | Yes, I need that it's non-empty. Thanks for pointing out. | |
| Nov 1, 2011 at 23:03 | comment | added | mhum | @Valerio Caprano: In property one, do you require that $[x,y]$ is non-empty for all $x,y$? If not, you could take $\Gamma = E$ for any $X = (V,E)$ which will satisfy properties one through three. I don't have a sense of what contractibility might mean in the case of graphs. | |
| Nov 1, 2011 at 21:52 | comment | added | mhum | I am mistaken. It turns out not to be the case that $\Gamma$ forms a tree. Consider $X$ to be a triangle with $\Gamma$ equal to the three edges. | |
| Nov 1, 2011 at 20:11 | comment | added | Valerio Capraro | Maybe this is true, but I am not sure. | |
| Nov 1, 2011 at 18:40 | comment | added | mhum | Ah, okay. In that case, would that imply that the union of all paths in $\Gamma$ forms a tree inside of $X$? Maybe I am still confused? | |
| Nov 1, 2011 at 18:12 | comment | added | Valerio Capraro | I don't think so, because $\Gamma$ might be a proper subset of $\mathcal C$. Think, for instance, at the unit ball: $\mathcal C$ is the set of continuous paths and so you have many ways to connect to points, but you choose just one: the segment line. Well, basically, what I am asking, is the existence of a choice of paths such that blablabla. | |
| Nov 1, 2011 at 17:50 | comment | added | mhum | I may have misunderstood your definition, but it seems to me that the first property implies that there is a unique path between any two vertices. If that is the case, then X is a tree, extremal vertices appear to be leaves in the tree, and the second property seems to imply that the graph is just a path. | |
| Nov 1, 2011 at 14:56 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 1, 2011 at 14:42 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
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| Nov 1, 2011 at 14:36 | history | asked | Valerio Capraro | CC BY-SA 3.0 |