Two men, aged 79 and 59, were hurt alongside a police officer who responded to reports of an incident in Moortown during the early hours of last Saturday.
It later emerged that the 79-year-old man, named locally as Ronnie Simpson, was bitten when he heard two dogs barking in his back garden on Garth Walk and went to investigate.
It is thought that the 59-year-old man, who also lives on the street, was then bitten after the dogs ran out of the garden.
He suffered were described as “potentially life-changing” injuries.
Two people, aged 17 and 44, were arrested by police and have since been released under investigation.
West Yorkshire Police today said that the owner of the two dogs had agreed with the decision to put them down, meaning a destruction order did not need to be sought through the courts.
The force also said that the breed of the animals had now been confirmed as American Bulldog, not American pit bull as suggested in the immediate aftermath of the attack.
A spokesman said: "They have been examined by police specialists and an independent expert and confirmed as American Bulldog, which is not a banned breed."