Golubic had little trouble in ousted fifth-seeded Kiki Bertens 6-2, 6-2 in 69 minutes. She won more than two-thirds of the points on serve and saved the only break point Bertens managed in the match. She also had a pair of breaks in each set.
Martic outlasted fourth-seeded Andrea Petkovic 6-7 (5-7), 7-5, 6-3 in a match that lasted 3 hours 11 minutes. There were a total of 14 breaks in the match, with Martic getting three of her eight in the final set.
No. 2-seeded Annika Beck got by Shahar Peer 7-5, 6-3 but needed seven breaks to do so and sixth-seeded Chanelle Scheepers also had seven breaks in her 6-1, 6-4 win over Anna Schmiedlova.
Mandy Minella posted a 6-2, 2-6, 6-3 win over Dias Evitmova and wild-card entrant Lisa-Maria Moser stopped Elena Bogdan 7-6 (8-6), 2-6, 6-4 in the only matches other than Martic's victory, on Tuesday's schedule to go three sets.
Other first-round winners Tuesday were Andrea Hlavackova, Patricia Mayr-Achleitner, Elina Svitolina, Estrella Cabeza Candela and Alexandra Cadantu.
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