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Berdych and Almagro received first-round byes, and had contrasting wins.
Almagro, a three-time finalist in Bastad, got off to a rough start against fellow Spaniard Guillermo Garcia-Lopez but rallied to win 3-6, 6-1, 6-1.
His next opponent, Fernando Verdasco, also of Spain, beat Jan-Lennard Struff in straight sets.
Berdych, the No. 1 seed in the clay-court tournament, easily defeated Argentine Martin Alund 6-3, 6-0.
It was the Czech's first match since losing to Novak Djokovic in the Wimbledon quarter-finals.
Next up for Berdych is Thiemo De Bakker, the Dutchman who survived two match points before eliminating Viktor Troicki of Serbia 4-6, 6-4, 7-6 (6).
Results from the Swedish Open (x denotes seeded player):
Second round:
Tomas Berdych (CZE x1) bt Martn Alund (ARG) 6-3, 6-0
Thiemo de Bakker (NED) bt Viktor Troicki (SRB x6) 6-4, 4-6, 7-6 (8/6)
Fernando Verdasco (ESP x8) bt Jan-Lennard Struff (GER) 6-1, 7-5
Nicolas Almagro (ESP x2) bt Guillermo Garcia-Lopez (ESP) 3-6, 6-1, 6-1
