I'm live on my own with my children, no partner. To me that's whats classed as a single parent.
When I was OD some guys would say they were 'single dads', yet on speaking to them further it turns out they don't live with their kids just see them as per arrangements with the mum.
There was also someone who has sole custody of his child but is living with a partner for a few years yet still calling themselves a 'single dad' ....to me that's curious as they are no longer 'single' and are very much parenting with their partner who has loads of input.
It rankles with me a bit to call yourself a single parent but have all the luxuries of a live-in relationship....