Definition talk:Vertex Deletion

I don't understand the need for the explains tag; as written the definition only applies to undirected graphs. If we want to extend this definition, we can, but currently it's just applying to graphs, not digraphs. Scshunt 23:46, 17 February 2012 (EST)
 * Good call - I have amended the definition so it specifically mentions that point. --prime mover 03:57, 18 February 2012 (EST)
 * I would say that a digraph should be a graph with extra structure; strictly formally this seems not to be the case at the moment, which would imply that there is no canonical forgetful functor from digraphs to graphs (i.e., a formal interpretation of 'forgetting' the direction of the arrows). However, I might be too influenced by axiomatic mathematics and rigorous set theory, and if this is the convention in literature (the paradoxical situation that a directed graph isn't a graph) it might be better to stick with it. Lastly, let it be noted that I generally haven't concerned myself with graph theory; it's just my perspective as a newcomer. --Lord_Farin 04:32, 18 February 2012 (EST)
 * At this stage of development it's useful to consider graphs (particularly not multigraphs) as a relation on a set of points: an undirected graph is equivalent to a symmetric relation, while a directed one loses this requirement. Alternatively, every undirected graph can be considered as a digraph with two arcs between each pair of connected vertices: one going one way and one going the other. So technically an undirected graph can be considered as a digraph with a constraint. Once we do functors and forgetfulness, your statement about that may become appropriate.
 * Mind you, I agree on this: in graph theory, while definitions can be appreciated in an intuitive way by describing their appearance in a depiction, in order to completely define an entity, we need to find a rigorous non-graphical definition, e.g. via matrices - but then the work of proving the various results becomes so much more unwieldy. --prime mover 06:04, 18 February 2012 (EST)
 * Matrices have the problem that they generally confine themselves to countable situations (mostly finite; the countable is already stretching it). I will try to concern my mind, which is as of yet unspoilt by convention, over it to try and think of an axiomatisation. --Lord_Farin 06:23, 18 February 2012 (EST)
 * There is no really natural way to associate them because the natural functor (association, mapping, whatever) from graphs to digraphs transforms an edge in a graph into a pair of edges in the corresponding digraph, one in each direction. But the underlying graph of a digraph contains an edge wherever there is an edge in either direction in the digraph.
 * It's also a much simpler treatment to go with edges as unordered pairs (sets of size 2). If you go with edges of a graph being defined in terms of a symmetric relation, then you can't really refer to "an edge" directly, and then any means you choose to refer to an edge is just really a more convenient definition of an edge. One other definition approach is similar to that of category theorists: a set of vertices and a set of edges, and a mapping from edges into V (in the case of a digraph, into V x V or two mappings, each into V). This seems like it would be more cumbersome than not, though.
 * (For the record, most of my knowledge in the area comes from courses at the University of Waterloo, not from books)