Talk to your insurance agent.Tell him you have your first rental and want a home owners policy tailored for this purpose.Every landlord I know who has been in the business for any length of time has a 1 million dollar umbrella policy.

@Michael Joseph I would recommend first calling whoever you use for your personal auto insurance to see if they offer umbrella policies.

Would umbrella liability insurance help in this case ?

We also would like to avoid them being named as an owner to prevent them from any liability (though an umbrella policy may be the solution to that).

But whatever, a huge majority on this site don't seem to care about that and will just suggest buying a massive umbrella policy. 

I have another homeowners policy with USAA on my primary, and an umbrella policy through on of USAA's intermediaries.

They trimmed the bushes off the roof of the house, bought a new mail box out of their own pocket.But I'm finding a few things that I'm not sure whether they are damage or not, and others that are clearly damage, but so minor that I'm wondering if it worth the hassle dinging their security deposit.Tenant is a married couple with two kids, though, they violated several terms of the lease:1) Got a dog [pit bull] that was not allowed. 2) Her sister and what appears to be one of her kids moved in with them at some point apparently after her divorce, I presume less than a year. 3) They got a pool that, along with the dog, caused me to have to get new insurance which was a huge hassle since I have 5 policies and an umbrella policy and have to move them all to get the best rates and the umbrella...With that background here come the questions:Obvious damage but not sure if I should ding them:-Two torn/missing blinds in front of house.

Buy an umbrella policy.

You could ask them to get a $500k umbrella policy with you as the covered insured...they will not want to do that and will probably drop it.  

My thinking surrounding the definition of a vertical model for a TKP would be a provider that maintains most or all of their provided services under the same roof or umbrella (e.g. they ultimately answer to the head of the same organization).