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February 25

3d analog of standard graph

This is hard to search for since you just get results for "planar graph". Can you have an analog of 2d graphs in 3d, using planes instead of lines? Each vertex would be a line segment, and the edges would be planes connecting two line segments. Does this have a name? Is it studied? 24.255.17.182 (talk) 03:05, 25 February 2016 (UTC)[reply]

See graph of a function. A function of two variables would give you a 3D graph. A function of even more variables would give you even higher-dimensional graphs. —SeekingAnswers (reply) 03:29, 25 February 2016 (UTC)[reply]

That is in no way relevant to my question. 24.255.17.182 (talk) 04:27, 25 February 2016 (UTC)[reply]

I'm having a little trouble following your question, to be honest. What is a "plane connecting two line segments"? Do the line segments have to be coplanar? If not, then how do you connect them by a plane? ~~If so, then doesn't the graph decompose into components with all edges coplanar, and in that case, in what sense is it three-dimensional?~~

Is it possible you're really interested in taking three vertices at a time, where an ordinary graph takes two at a time? If that's what you're getting at, see hypergraph. --Trovatore (talk) 04:42, 25 February 2016 (UTC)[reply]

Update: I realized the struck-out part is wrong — you can have a set of line segments that are pairwise coplanar without there being a single plane containing all of them. Is that what you were getting at? I sort of doubt it but I'm not quite sure. --Trovatore (talk) 04:55, 25 February 2016 (UTC) [reply]

Do you mean to start with a 2D graph, like this:

+ 
| 
| 
|     
+-----+
|      \ 
0-------+---+

And make it 3D, like this:

   +
  /|
 / |
+  |
|  +-----+
| /     / \
|/     /   +---+
+-----+   /   /
|      \ /   /
0-------+---+

Is that what you mean ? StuRat (talk) 05:58, 25 February 2016 (UTC)[reply]

You know, that's exactly what I thought OP meant, and also some sweet graphics :) We don't have an article for ribbon graph, but that term gives us lots of similar images via google [1]. Each vertex in the 2d graph becomes a line segment in the 3d ribbon, and each line segment in the original graph becomes a plane joining two line segments. SemanticMantis (talk) 20:33, 26 February 2016 (UTC)[reply]

It sounds like you're asking about graphs that can be embedded in 3-space by sending vertices to line segments and edges to planar quadrilaterals. I'm fairly certain that these are precisely the graphs that can be embedded in 3-space using the standard sort of embedding. Given a standard embedding in 3-space, wlog no two vertices share a z-coordinate. Now replace every vertex with a very tiny vertical line segment and thicken each edge to the appropriate quadrilateral.

As for which graphs can be embedded in 3-space in the standard sense, precisely those with only countably many vertices.--2406:E006:786:1:7DEC:7344:2FA3:4661 (talk) 11:22, 25 February 2016 (UTC)[reply]

I'm curious about whether there exist finite graphs whose vertices cannot be assigned 3space coordinates such that the resulting Delaunay graph has a subset isomorphic to the original. —Tamfang (talk) 00:35, 28 February 2016 (UTC)[reply]

Possibly what the OP is looking for is Polyhedron and Platonic solid. Instead of being bounded by lines, these are bounded by planes. Loraof (talk) 14:11, 25 February 2016 (UTC)[reply]

I think you want a general form of Graph embedding, so that would involve embedding simplices - see Simplicial homology. Planar graphs are graphs embedded into a plane. You want surfaces linked along their edges and to embed them into 3D space. With straight lines you'd be talking about things like rigid origami or mechanical linkages. Dmcq (talk) 14:32, 25 February 2016 (UTC)[reply]

Another possibility is Abstract polytope. Asking for the X analogy of Y is generally somewhat vague; if you can find a precise definition or have an application in mind it would be better. --RDBury (talk) 16:08, 25 February 2016 (UTC)[reply]