NEW DELHI-—The leaders of India's Congress party, Sonia Gandhi and her son Rahul, offered on Monday to resign after the party's most dismal election performance ever, but the party refused to let them go.

The Congress party—which has governed India for the last 10 years and most of the country's modern history, led by the Nehru-Gandhi family—secured just 44 of 545 seats in India's lower house of Parliament, while its main opponent, the Bharatiya Janata Party, won a landslide victory with 282 seats.

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