Did they have their home burned, too?
This headline over an AP story graced my local paper:
had throats slashed
The lede had the same problem:
Three children found dead in the ruins of their burned home had their throats slashed before one the parents set the fire, authorities said Tuesday.This is a problem called volition or false purpose (for a discussion, see Kansas Professor Malcolm Gibson's fine editing site). It's not always easily solved. Change it to "had slashed throats" and there's still ambiguity -- did they slash their throats before the fire? It can be read that way if someone mentally slips in the word "their" between "slashed" and "throats."
The real problem here is that leading with the children makes them actors, when they were passive receivers. This might be a case where passive form in the hed works better:
slashed before fire
The throats of three children found dead in the ruins of their burned home were slashed before one of their parents set the fire, authorities said Tuesday.