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Subsequently, Graf terminated his study of architecture in order to devote himself entirely to music. Aside from his studies with Seidlhofer he took counterpoint and composition classes with Alfred Uhl. 1949 he passed the First State Examination and became a finalist in the 1950 Geneva International Music Competition. Having also passed the Diploma Examination in piano in 1951, Graf gave his debut that same year as a soloist at Konzerthaus, Vienna. In the following year he performed several concerts: 1952 at the 4th Queen Elisabeth Music Competition in Brussels, 1953 at the very first International ARD Music Competition in Munich, 1954 he participated for the second time in the Geneva International Music Competition, where he was awarded second place, and again in Brussels at the 1956 Queen Elisabeth Music Competition, where was among the finalists along with Vladimir Ashkenazy, Lazar Berman, André Tchaikowsky, John Browning and Tamás Vasary.