Strict liability for the owner, keeper or harborer
Under R.C. 955.28(B), the owner, keeper or harborer of a dog is liable in damages for any injury, death or loss to person or property that the dog causes, unless the injured person was committing a crime on the property or was teasing, tormenting or abusing the dog.
A harborer is one who actively shelters, protects or exercises control over a dog; owning the premises is not enough. The two-year personal injury deadline is the safe assumption.