The friend probably has no idea she's 'rudely dominating the household'. People tend to not realize this unless they are told.
I was in a similar situation once, resented two very lively dogs being brought over to tear up my yard and use it as a toilet. My dog was out in the yard, but until her dogs came over and started chasing him, he didn't do any harm. All of them, tearing around at 140 mph for 4 hours, DEFINITELY did some damage. If I put mine up, her two, all on their own, did a rather amazing amount of excavating.
How it started, actually, was that a gal who was pet sitting, started bringing her dogs over...at another home of ours, it was where a friend of mine invited a group of people over, with my blessing, but unbeknownst to me, told them since it was a farm, it was the ideal place for them to all bring their dogs and let them loose to 'play'.
There were more big dogs there than at the screen tests for 'Marmaduke'.
I can go on and on about how wrong that is, and how unfair, and how my flowers in my front yard are just as expensive as those in a small yard, and all, but the fact is, every single time she came over, I pretended, I slapped a smile on my face, and let it go on, and my friend had no idea about the seething bubbling lava under that calm exterior...my SO said it was a matter of having the courage to speak up...or not. He said keeping quiet about it was 'dirty pool' on my part, and he was right.
People are not mind readers. Most people have no idea when they're being rude or about all that lava bubbling beneath the surface, especially when, as is usually the case, no one tells them...or tells them clearly, honestly and completely.
No, I am not saying you don't like the dog - that was more for the folks advocating putting him in with the big dogs and if he gets bit up, 'oh well'. So were the comments on it's the owner, not the dog, as other posts seemed to be suggesting it was somehow incumbent on the DOG to learn to poo here and not there.
If you tell them, tell them politely. 'I am sorry, but your dog can't come over any more'.
Of course, it often goes like this.......'WHY?' 'Because this is the tenth pile of poop he left in front of the tv'. 'That's not my dog's poop'. 'My dogs are Bulgarian Bear Dogs, and weigh 250 lbs. Yours is a chihuaha. That is your dog's poop'.
But try to keep it as level headed as you can. 'You are welcome here but your dog is not' never goes over too hot, no matter who it is. Mine was as good as it gets, 'I really do love your dogs, they are lovely dogs, but please, the next time you come over, please leave them at home'.
The fact is, most dog owners just aren't very sensitive to how other people feel - most people in general aren't either. They just aren't. They tend to see things from their side, not the other side. They need a wee it of guidance in this. To boot, many dog owners are very pressed for time, and really need a way to get the dog some exercise and play. Many are too tired (or lazy) to drive the dog to the dog park, or take it for a bike ride, or put it on their treadmill at home regularly. If they take a dog someplace and the person there doesn't complain about it, it WILL continue.
Then you get to find out if you are a toilet break and a rest stop, or a friend. Most of the time, especially if you have some land, it will be the former. Whenn I told the lady with the two dogs to leave the dogs home when she visited, that was the last time I ever saw her at my house, got a phone call from her, or had an email from her.
On the other hand, I heard about it for a very long time from others, and that was very, very not pretty, she actually got it so the whole group got mad and stopped coming over. So you have to be prepared - when you speak up, no matter how gently, there usually is fallout. You just have to decide what is more important to you.
Her side was I HAD a dog, and it was loose in the yard ALL THE TIME, so what's the difference if there are a couple more out there? I was blaming her dogs for what my dog was doing, and it got much much worse from there.
So it's good to know, that even if you say it very politely, you can very well lose a friend over it. But anyway, it's good to know. One doesn't need 'friends' like that.