Dear Bob
I've been following the recent posts on undirected graphs with interest. But I have a question. I think it's being said that undirected graphs are the same as directed graphs with involution. (Presheaves on the full subcategory of SET determined by 1 or 2, or just 2.) Which is nice but what about loops? The involution might fix a loop or not. So wouldn't we be getting undirected graphs with two kinds of loops, whole loops and semiloops? What am I missing?
There has been some 'work in progress' at Bangor on this very question for the past 8 years. This is intended to present a categorical approach to graph theory to workers in combinatorics, and is not intended for category theorists. The current draft, 06.04, is available at http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/06... (follow the link 06.04 to 06_04.pdf). We do indeed discuss two types of loop, which we call 'loops' and 'bands'. Ronnie is away today, but may well add his own comment tomorrow. Best wishes, Chris Wensley