Is there sin incurred by the father or the son when killing a dog with a broken leg by injection, after the veterinarian determined that there was no benefit in treating it and that it would suffer and die?

A Muslim is not permitted to keep a dog unless it is a hunting or guard dog; otherwise, a Qirat of his reward will be deducted every day. As for killing a dog, it is not permissible except for a reason, and there is no sin upon you in killing it if the veterinarian deems it necessary.