Anti-Border Corruption Act of 2010 - Requires the Secretary of Homeland Security (DHS) to ensure that: (1) by not later than two years after enactment of this Act, all applicants for law enforcement positions with U.S. Customs and Border Protection (CBP) receive polygraph examinations before being hired for law enforcement positions; and (2) by not later than 180 days after enactment of this Act, CBP initiates all periodic background reinvestigations for all of its law enforcement personnel. Requires the Secretary to make periodic progress reports to Congress on CBP compliance.