Definition talk:Edge Contraction

A few notes here from my first draft:

The notation for the vertex set of $H$ is a bit dense, but the other alternative is longer (since you need to handle the separate vertices of $e$ separately) and leads you to need more vertices explicitly named.

A common trick is to let the new vertex be simply $e$ itself, but under our current definitions, edges and vertices need not be disjoint. --Scshunt

Yes, I was going to question the notation on this page. I understand that what you have used may be neat and compact, but the following are not clear and need to be explained:

a) $V\left({G}\right) \setminus e$ ... we have that $V\left({G}\right)$ is a set of vertices, and $e$ is an edge.

i) For a start, I would begin by writing $V\left({G}\right) \setminus \left\{{e}\right\}$ as strictly speaking (particularly as this site has no current endorsement of that particular notation abuse) you can't use a set operator on an element.

ii) ... and further, as $V\left({G}\right)$ is a set of vertices, then you can't really subtract an edge from it.

I have no philosophical problem with the notation, as long as it is explained. So, what I suggest is a page which specifically explains that $V\left({G}\right) \setminus e$ means (as I presume): "The set of vertices of $G$ from which the vertices at either end of $e$ have been taken away", or some such. Then link to that page.

b) $\left\{{f \in E\left({G}\right) : f \cap e = \varnothing}\right\}$ also a bit obscure. Again, a page is needed defining $f \cap e$ which from what I gather means "the vertices incident to both $f$ and $e$.

c) $f \setminus e$ is, I would guess, the set of vertices adjacent to $f$ which are not also adjacent to $e$.

Don't get me wrong - I like the compactness of this notation - but the philosophy of this site is that each of these notations needs to be explained.

I also think a diagram would be useful, with a "before" and "after". I'll see if I can get it together to craft one.

I also think it worth illustrating what happens when the edge is a loop. I think it means just that the loop itself ends up with being removed.

BTW: Pls remember to sign your posts - use the "squiggle" icon above the edit pane, 3rd from right, which adds 2 hyphens and 4 tildes, which gives your sig and a timestamp. --prime mover 03:36, 17 March 2012 (EDT)

Had another thought: how about defining an operator $v(e)$ on an edge $e$ meaning "the vertices adjacent to $e$, and /or $v(\{e, f\})$ meaning "the vertices adjacent to $e$ and $f$", and can be expanded to take any subset of $E$? Call it the "adjacency operator" or whatever, and define something similar for vertices? Or has this already been considered in the world of graph theory (with which I am out of touch)? --prime mover 03:40, 17 March 2012 (EDT)

This comes back to the problem of definitions, which we've discussed before. The definition of (simple) graph we use defines an edge as a 2-set of vertices, so the three notations I used are applying to edges as sets. A diagram would be useful. I don't know about looped graphs; the definition using 2-sets implicitly does not allow loops. Scshunt 18:33, 18 March 2012 (EDT)