Petra Kvitova came from a set down to beat Katarzyna Piter on her return from injury at the Bad Homburg Open, while no play was possible in Eastbourne on Monday due to rain.
Kvitova withdrew from the French Open after suffering a freak ankle injury during her post-match media duties at Roland Garros.
The two-time Wimbledon champion recovered to get her grass-court season under way in Germany and battled to a 4-6 6-1 6-4 over Polish outsider Piter.
A week before the third grand slam of the year gets started at the All England Club, top seed Kvitova trailed 2-0 in the final set as she struggled with her serve, but broke three times to book a second-round meeting with Ann Li.
Sloane Stephens withdrew ahead of her scheduled match with Laura Siegemund due to a foot injury. Riya Bhatia stepped in with Stephens absent, but Siegemund consigned her to a 2-6 7-6 (7-1) 6-1 defeat.
Angelique Kerber needed only 50 minutes to see the back of Ekaterina Yashina 6-1 6-1 on home soil, grasping all five break-point opportunities.
Kerber will do battle with Anna Blinkova in round two and Jessie Pegula's next assignment is an encounter with Katerina Siniakova following a 6-1 6-2 defeat of Amandine Hesse.
Blinkova beat Clara Tauson 7-6 (7-4) 4-6 6-2, while Sara Sorribes Tormo saw off Martina Trevisan 6-4 3-6 6-2
Miserable weather in Eastbourne prevented Karolina Pliskova, Madison Keys, Bianca Andreescu and Madison Keys from playing at the Viking International.
World number one Swiatek sees off Xiyu in Madrid opener
World number one Iga Swiatek overcame Wang Xiyu with the minimum of fuss on Thursday to progress to the third round of the Madrid Open.
Swiatek has yet to win the title in Madrid, which is the only clay-court European event at which the Pole has not triumphed, but she got off to a flying start with a 6-1 6-4 victory over Xiyu.
"I love this place. I got to know the city a little better last year," Swiatek said. "So this time I feel more comfortable around."
Next up for the 22-year-old is a third-round tie against 27th seed Sorana Cirstea on Saturday.
Data Debrief
Swiatek has registered 150 match wins on the WTA Tour since the start of 2022. Elena Rybakina and Coco Gauff are the next best with 108 wins each.
She also became the first player born this century to claim 50 combined match wins in Madrid, Rome and Roland-Garros, since the inaugural women's singles event at the Madrid Open in 2009.
Dominant Gauff double bagels Rus at Madrid Open
Coco Gauff made a flying start to her Madrid Open campaign on Thursday, recording the first double bagel main-draw win of her career as she cruised past Arantxa Rus to reach the round of 32.
Gauff needed just 51 minutes to see off 33-year-old Rus as she booked a meeting with Ukraine's Dayana Yamstremska for Saturday.
The US Open champion lost just 18 points throughout the match – eight in the opener and 10 in the second set – as she put any doubts over her recent form to bed in emphatic fashion.
Gauff's power proved too much for Rus in the opener, the third seed winning 88 per cent of points behind her first serve and converting three of four break points to go a set up in just 23 minutes.
Rus brought up four break points midway through the second set, but when she failed to convert them, Gauff punished her in ruthless fashion by reeling off another three games then finishing things with a terrific forehand winner.
Speaking after her win, the American said: "I played really well today. It was just one of those days for me… 6-0 6-0, I've never done that before, I'll probably never do it again."
Data Debrief: Gauff matches Azarenka, Halep feat
Gauff is just the third player to register a 6-0 6-0 win in the women's singles draw at the Madrid Open, following in the footsteps of two former world number ones in Victoria Azarenka and Simona Halep.
The American, who only turned 20 last month, is also the youngest player to double bagel an opponent in any main-draw match on the WTA Tour since Iga Swiatek did so against Karolina Pliskova in the 2021 Italian Open final in Rome.
Nadal cruises past teen Blanch to tee up De Minaur rematch in Madrid
Darwin Blanch presented little challenge for Rafael Nadal, who dropped just one game in a convincing 6-1 6-0 triumph at the Madrid Open.
Nadal is feeling his way back to fitness, but 16-year-old qualifier Blanch offered little resistance to the 22-time major champion on Thursday.
Making his farewell appearance at the Madrid Open, where he has won five titles, Nadal needed just 64 minutes to progress to round two.
"I think today I played against an opponent with a great future in front [of him], but today, still making mistakes," said Nadal.
"I just tried to be there, be solid all the time without taking a lot of risks. It worked well. I'm happy to be through and I wish him all the very best for the future."
Nadal's reward is a rematch with Alex de Minaur, who he lost to last week in Barcelona.
Data Debrief
This game saw the largest age gap (21 years, 117 days) between two opponents in the history of ATP Masters 1000 events.
Nadal has now registered 57 wins at the Madrid Open, which is 20 clear of any other player.
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