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Nine people were killed when a suicide bomber blew up a belt of explosives at an entrance to a parking lot in the Shia city of Karbala, 110 km south of Baghdad, Efe news reported.
A few hours later, 20 people were killed in the second suicide attack on a market in the city of Musayyib, also located south of the capital, the Interior Ministry said.
At least 21 were injured and several shops, stalls and nearby cars were damaged in the second bombing.
The Islamic State terror group claimed the attacks in a statement on its Amaq website and said the operations were aimed against Shia groups in Karbala and Musayyib.
The attack came as Iraqi security forces, backed by the anti-IS international coalition, were simultaneously conducting a major offensive to dislodge IS militants from their remaining redoubt in the western side of Mosul.