Free interpolation in Holder classes (with exponent < 1) is studied for domains of the complex plane under the assumption that the length of an arc on the boundary of the domain is commensurable with the length of the spanning chord. A subset of a domain on which the restriction of a Hölder class over the domain is the corresponding Hölder class over the subset is considered and its structure is shown. It is also proved that the description of interpolation sets is similar to that for a disk.