Immigration documentation should provide you w/exact place of origin in Ukraine. The Australian Archives are free to citizens and available online. All DP's/Displaced People immigrating to Australia post WW2 had to fill out extensive personal questionnaires. These records should confirm all kinds of details. You might find that family is listed as Polish. Don't let that throw you. For a period of time post WW2, the Allies did not recognize Ukrainians as a nationality, so they lumped them in with other ethnic groups, depending on the area they were originally from. As per personal questionnaire, you should also be able to get a handle on the demczyk/demczuk and ziemlaneyj WW2 experience and the exact place of origin.
National Archives of Australia - The Collection - Family History - Immigrantshttp://www.naa.gov.au/The_Collection/Family_History/immigran...
ISTG - World War II Refugees to Australiahttp://istg.rootsweb.com/ww2_au.html
above link excerpt from
Shipping and Immigrationhttp://www.coraweb.com.au/shipsite.htm
DP Camps in Europehttp://home.earthlink.net/~okaczmar/dpcampseurope.html
World War II in Ukrainehttp://www.infoukes.com/history/ww2/
Ukrainian History: Chronological Tablehttp://ukraine.uazone.net/history.html