Visit to a different university - is it OK to bring my girlfriend along? I am a PhD student, and I was recently invited to visit a university which happens to be at a very attractive touristic location. 

It goes without saying that the purpose of the visit is purely scientific, and any other matters are  tangential. On the other hand, it has occurred to me and my long-time girlfriend that it would be a perfect opportunity for her to see this touristically attractive place. Now, I am wondering how best to make the arrangements, so as to maximize both research and personal benefits (and not offend the host as a by-product).

To state the obvious, at the time when I am supposed to be doing research I will be doing research, and not get distracted by the presence of my partner. The place is interesting enough to keep her entertained through the day, and to keep us both entertained during the evenings. I would not expect the host to have much interest in how I spend my free time if I want to be left to my own devices, so if me and my girlfriend make totally separate arrangements and only spend the evenings together, this should not cause any problems.

Keeping things entirely separate feels a little impractical though, so I have some questions about issues that I do not see as so clear-cut. My host will be booking accommodation for me and paying for it. Is it impolite to suggest that he book a double room, and we pay half of the price from our own pocket? It is probable that I will be invited to some social gatherings - maybe beer after a seminar, maybe dinner. Is it polite to bring my partner along/resign from joining? Would it be OK to streach the visit for an extra weekend after the research is done (paying for the hotel, of course, but hopefully still getting the return flight refunded)?

I would also be grateful for input on any issues that I have not yet thought of.

A few details, just in case they are relevant: My field is (pure) mathematics. The university is in Europe, at the Mediterranean coast. The visit is about a week long. We are in a long-distance relationship, so a extra week together does make a difference. The host strikes me as rather easy-going, but I don't know him that well.
It is perfectly reasonable. I have done it a couple of times and I know colleagues that frequently do it. Her presence at work/social events depends on the event and how well you know the host and the events. Often the work/social line gets blurred. I am friends with a number of my colleagues partners and my partner are friends with them and their partners. Sometimes my partner will join me for social events when we have a visitor.

That said, while it is not inappropriate, the trailing partner at academic events never has worked well for us. I have been both the leading and trailing partner. The person working is often tired at the end of the day and not really in the mood to be a tourist. We find it is a better use of our time and money to travel both as tourists and really enjoy our holidays.