As per Ellis Island, there is an Iwan>later anglicized to John Dobrowolsky, 25 yrs old (YOB 1881???)>, single, ethnically listing himself as Rusniak/LEMKO arriving from Lezovate/LESZCZOWATE May 6, 1906, heading for Carteret, NJ to meet up w/brother Fedko.
Lemko village Leszczowate (coordinates 49.30-22.34, very close to what is Ukrainian/Polish border today> Lisko was administrative district and Uztrzyski Dolne was judicial/tax district for the village. The Greek Catholic Church was in Leszczowate and the filial Roman Catholic Church was in Jasien. This region was Western Galicia>administered by ethnic Poles for the Austro-Hungarian Empire. It was also frequently referred to as Lemkivschyna. Today this is Podkarpackie Region in SE POLAND.
Central Europe Review - Karen M Laun: The Lemko of Poland http://www.ce-review.org/99/24/laun24.html
LEMKOs | Lemkivshchyna | Lemkovyna | Lemko | Boyko | Rusyn | Rusnak | Ruthenians | Carpatho-Ruthenians, etc..... http://www.lemko.org/
I AM NOT SURE THAT THE ABOVE IWAN IS YOUR GGRANDFATHER. He was the only Iwan that was heading for Carteret. You should also check every JAN Dobrowolski/y as well. Look in the Ellis Island original ship manifests/handwritten copies for a Jan (Polish name for John) heading for Carteret.
Once you have established which Dobrowolski/y was your own, check CHURCH RECORDS for Grandfather's birth records, GGrandfather's marriage records etc. Where was GGrandfather buried? There is an extensive library in Carteret. AND DON'T NEGLECT DOCUMENTING GGRANDMOTHER & GRANDMOTHER.
Before crossing the Atlantic you have a lot of work ahead of you in States.
PS> Technically the best way to research your ancestors is to document you own parents and then keep moving backwards and documenting. Documentation will move you from generation to generation. Lemkos stuck pretty close to their own.