Talk:Incidence structure

An incidence structure is not a concept belonging to combinatorial mathematics unless the sets involved are finite (from this point of view, graphs and hypergraphs with an infinite number of vertices are not combinatorial objects either.) The only reference given clearly has a bias toward calling things combinatorial. Incidence structures arise in geometry, but they are more general than that subject. There is no particular reason to place this concept in any subtopic of mathematics (Combinatorics, Geometry, Design Theory, etc.). We are simply talking about two sets with a symmetric relation between them, a structure that can come up in any discipline (and saying that it "belongs" to Set Theory, just doesn't say anything to most readers). Wcherowi (talk) 04:46, 1 September 2011 (UTC)[reply]

Incidence structure vs Incidence geometry[edit]

Can someone clear up the distinction between Incidence structure and Incidence geometry? 195.77.88.65 (talk) 20:46, 15 April 2014 (UTC)[reply]

One of them is a field of study, and the other is the thing being studied. —David Eppstein (talk) 21:07, 15 April 2014 (UTC)[reply]

generalization[edit]

What about to arbitrary dimension?79.180.20.102 (talk) 16:29, 8 October 2014 (UTC)[reply]

The definition of an incidence structure is so general that the concept of dimension may not apply. Points and lines in an incidence structure are primitives (they have no definition) and all other structures, such as planes, solids and more general sets are defined in terms of these primitives. This can be done in several ways, not all of which lead to well defined concepts of dimension. In this general setting you can define dimension in the following way: First you define the different types of objects that you are dealing with (this set of things are called points, this set of things are called planes, etc.) and when an object of one type is "incident" with another (when is a point in a plane? when is a line in a plane? etc.) Then you define a flag as an ordered set of objects, at most one from each type, so that each object is incident with the next object. A maximal flag is one that can not be made any larger. If all the maximal flags have the same number of objects then this number is the dimension of your incidence structure. I have taken some liberties in this description and not everyone would agree with the way I have used certain terms, but I hope you get a glimmer of what is involved. Bill Cherowitzo (talk) 17:40, 8 October 2014 (UTC)[reply]

Comparison with other kinds of structures.[edit]

The article now states

An incidence figure (that is, a depiction of an incidence structure), may look like a graph, but in a graph an edge has just two endpoints (beyond a vertex a new edge starts), while a line in an incidence structure can be incident to more than two points. In fact, incidence structures are hypergraphs.

This surprises me a bit.

Actually, the author of the sentence seems to take for granted that the only natural way to consider an incidence structure as a graph is by letting the lines in the structure somehow correspond to the edges in the graph. On the other hand, when e.g. there is a reference in Cage (graph theory)#Known cages to some cages as "incidence graphs", this implicitly treats the point set and the line set as the two parts in a bipartite graph, whose edges are given by the incidences. I would have expected a reference to this point of view here (if for no other reason to serve as a more adequate direct reference from the cages article).

Does anybody see a good reason not to rewrite the section, making the bipartite graph the principal example, and demoting the hypergraph interpretation to a secondary one? JoergenB (talk) 11:54, 25 May 2015 (UTC)[reply]

These are two different ideas, not just two ways to look at the same thing. However, this sentence could use some improvement (I'm not sure what to make of the remark that "beyond a vertex a new edge starts"). The incidence graph of an incidence structure is not a diagram of that structure, it is a visualization of the incidence relation defined by that structure (i.e., its Levi graph). On the other hand, the incidence structure is a hypergraph (actually, I like to put it the other way; every hypergraph is an incidence structure after the graph theorists have monkeyed around with the terminology.) The intent of the sentence seems to be that a diagram of an incidence structure "looks like a graph" but really isn't one. The same can be said of any diagram composed of points and lines joining some of them, so I'm not sure why incidence structures are being singled out here. In looking at the article as a whole, this section is clearly out of place and needs to be further down in the article (if it's going to be kept at all). I think some examples (other than the Fano plane) would be appropriate at that location, maybe something that is not uniform. The section on hypergraphs should be expanded to include the alternate terminology of block designs. If no one else wants to do these things, I'll be able to get back to this article in a few days. Bill Cherowitzo (talk) 17:41, 25 May 2015 (UTC)[reply]

Done But it could still use a little more polishing. Bill Cherowitzo (talk) 04:49, 30 May 2015 (UTC)[reply]

I don't think this is quite accurate. The Levi graph of an incidence structure is still an abstract mathematical object, not a visualization. It is only when one produces a drawing of this graph that one has a visualization. —David Eppstein (talk) 18:13, 25 May 2015 (UTC)[reply]

I stand corrected. I was being a little (too!) loose with my language since the technical verbiage was getting a bit too convoluted. Bill Cherowitzo (talk) 03:30, 26 May 2015 (UTC)[reply]

My suggestion partly was based on two misunderstandings; however, rewriting might make the risks for such misunderstandings smaller, I hope.

I thought that the main purpose of the section Comparison with other kinds of structure was to provide comparison of the incidence structures with other kinds of structures, and that this was supposed to be the leading part of the article with this purpose. I now realise that the purpose more may have been to warn readers early on not to confuse a common graphic representation of e.g. finite projective planes with the common graphic representation of graphs. Thus, I fear I was misled by the section title. A more adequate title (if the content and location of the section remain unchanged) might be "What the incidence structure is not":-).
A more encyclopaedic solution would be to rename it to Common illustrations of incidense structures or something similar, expand it a bit by describing and examplifying how an abstract incidence structure may be depicted, and then rewrite the rest as a warning to the readers that the similarity with graphic illustrations of graphs should not lead to the mistake of considering them as equal.
I searched the article content for references to incidence graphs, and found none. Obviously, I did not read the section on Levi graphs (or search the text for "incidence graph", which I should have done). I must confess that this is the first time I have heard the term "Levi graph" used as a synonym for incidence graph. I am not at all sure that it is a good idea to use the term Levi graph rather than incidence graph as the main term for that concept (especially as Levi graph also seems to be used for just one particular incidence graph, the (3,8)-cage; cf. [1]); but this perhaps concerns the article Levi graph more than this article.

As for structures, I still think that the incidence (or Levi) graphs are the main ways to represent incidence structures as (any kind of) graphs, and that the two representations as multihypergraphs (namely, with either collections of points or collections of lines as hyperedges) ought to come later. However, it might be better to mention the vertex-edge incidence system of a given (undirected multihyper)graph as an example. This also would make it natural to apply incidence counting for such a result as the theorem that there is an even number of odd degree vertices in a graph. JoergenB (talk) 18:53, 26 May 2015 (UTC)[reply]

Are we not missing some rules in the definitions?[edit]

In Incidence structure#Formal definition and terminology I think we are missing some rules.

As it stands at the moment all of the following are possible:

a: two lines sharing the same set of points

b: three points being part of two different lines

c: two points not having a connecting line

and maybe even more "strange" incidence possibilities. (add your own if you like)

If these possibilities are indeed possible I think they should be mentioned and I would like to have a (sub) section "related subjects" that links to structures / geometry / subject have a more limited definition.

Unfortunedly I am not knowledgable enough in this subject to write about it myself, also maybe different authors will have different definitions, but lets work on a wikipedia definition here :) WillemienH (talk) 19:08, 19 December 2015 (UTC)[reply]

Yes, those are incidence structures and in some situations they may even be useful examples. As they are not even partial linear spaces, they aren't really studied and I don't see a need to single these things out since they aren't particularly notable. Bill Cherowitzo (talk) 20:44, 19 December 2015 (UTC)[reply]

I would then suggest to reorder the article, I want to move the examples down because they all just represent a small subset of all possibilities (while other parts like dual structure and representation seems applicable to all incidence structures. WillemienH (talk) 22:24, 19 December 2015 (UTC)[reply]

Incidence complexes[edit]

Incidence complex is currently a red link. Should it be a redirect here as a synonym or subtopic, or is it a distinct object of study in its own right? — Cheers, Steelpillow (Talk) 08:58, 28 September 2022 (UTC)[reply]