First seem to be a bit like the Stagecoach in Newcastle down in York, they seem to do quite a bit of cross-city routes as stated by Sean. Transdev are also prevalent there, doing the long-distance routes to Scarborough, Leeds, and I think one to Whitby as well? Arriva don't seem to have as much of a presence down there, but I do see a few photos on Flickr that show they have a bit of ground in the city.

As also said, East Yorkshire Motor Services do long distance services too, I think there are a few over to East Yorkshire (duh!) to places such as Hull, Beverley(?) and maybe Scarborough as well? Again, as was said in a few posts above, they seem to use mainly Gemini 2s (although I have seen photos on Flickr of Gemini 1s - unsure if they are still in the city centre or have moved on to other work).