https://www.bbc.co.uk/news/amp/world-europe-41280080
The Bright Future party announced it was withdrawing from the three-party coalition after nine months in office.
It blamed a "serious breach of trust within the government".
Earlier it emerged that the prime minister's father had written a letter recommending a convicted paedophile have his "honour restored".
This old Icelandic system permits convicts to have certain civil rights restored - enabling them to run for public office, qualify for certain government jobs or serve as a attorney or solicitor, for example - if three letters of recommendation from "respectable" citizens are provided.
But Icelanders have been horrified by the secret backing for Hjalti Sigurjón Hauksson - convicted of raping his stepdaughter almost every day for 12 years.
The Rekjavik Grapevine news site quotes the survivor of Hauksson's abuse as saying it was "surreal" that he should receive restored honour.
The government has also been accused of an attempted cover-up after it refused to disclose who had written the letter of recommendation.
It only emerged on Thursday that it was Benedikt Sveinsson, Prime Minister Benediktsson's father, but the prime minister is said to have been informed about his involvement in July.
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Calls are growing for a general election as there are few possible options for an alternative coalition to take power, he says.
"Since the economic crash of 2008 there has been a lot of suspicion towards politicians and a huge onus on honesty," Gudmundsson says.
"This is not the first time controversy has been linked to the PM - he wasimplicated in the Panama Papers [which brought down the previous government, in which he was finance minister]. There isn't much public patience for this kind of thing."
[....]
Mr Sveinsson apologised for writing the letter of recommendation, saying that Hauksson had brought him an already drafted letter which he signed.
"I have never considered the restored honour as anything except a legal procedure making it possible for convicted criminals to regain some civil rights," Mr Sveinsson was quoted as saying by the Iceland Monitor .
"I did not think of it as something that would justify Hjalti's position towards his victim. I told Hjalti to face his action and to repent."
"What was supposed to be a small gesture of good will towards a convicted criminal has instead turned into a continuation of the tragedy for the victim. For this I again apologise."