Tarjan was just 17 when he was selected to the American team for the 1969 World Students' Olympiad, at Dresden. He was a member of the winning American side at Haifa 1970, and was selected again at Graz 1972.
He earned his International Master title in 1974, and was promoted to Grandmaster in 1976. He played for the American team at five straight chess Olympiads. He began at Nice 1974, then played at Haifa 1976, Buenos Aires 1978, Lucerne 1982, and Valletta 1980.
His best results in international tournaments include first at Subotica 1975, first at Vancouver 1976, and first equal at Vršac 1983, along with Predrag Nikolić and Georgy Agzamov. Other good finishes included tied for third at Chicago 1973 with 7/11; tied for fifth at Venice 1974 with 7.5/13; and an excellent tied second at Bogotá 1979, with 10.5/14, behind only Alexander Beliavsky.
Tarjan played in several U.S. Championships during the 1970s and 1980s. He was fourth at El Paso 1973 with 7.5/12. At Oberlin 1975, he ended up tied for sixth with a score of 6.5/13. At Pasadena 1978, which was the Zonal qualifier, he tied for second with 10.5/14, and advanced to the 1979 Riga Interzonal, part of the World Championship cycle. He scored 8.5/17 and did not advance; the tournament was won by former World Champion Mikhail Tal. Tarjan's last competitive tournament for three decades was the 1984 U.S. Championship at Berkeley, where he went out in style with a fine tied third, scoring 10.5/17.
In 1984, Tarjan gave up professional chess to become a librarian. He is a graduate of the University of California, Berkeley, and received his MLS from UCLA.
In 2014, Tarjan surprisingly emerged to play in the U.S. Open in Orlando. He finished with 7.0/10, half a point behind a six-way tie for first place.