Former New York Mayor Rudolph Giuliani was dragged into the swirling waters of Balkan politics on Friday after participating in public events with two opposition-party candidates in hotly contested election fights in Serbia. A senior politician from the Serbian Progressive Party, Aleksandar Vucic, who is running for mayor of Belgrade, said he invited Mr. Giuliani to advise him on how to revitalize the Serbian capital on the Danube River and pull it from an economic slump. Mr. Giuliani, who ran for the Republican presidential nomination in 2008 and now works as a consultant, also met with Tomislav Nikolic, the party’s candidate for president, in what is expected to be a close race against the pro-Western incumbent, Boris Tadic, of the Democrat Party. Messrs. Vucic and Nikolic used to be prominent members of a staunchly nationalistic party that opposed European Union membership and condemned U.S. involvement in the Balkans. They left in 2008 and launched the Progressive Party, which says it supports Serbia’s eventual membership in the EU.