HE BECAME known as the Facebook killer because it was on the social networking site that he broadcast his intention to kill his daughter: ''Bout 2 kill ma kid,'' wrote Ramazan Acar shortly before he murdered two-year-old Yazmina in Melbourne in November 2010 by stabbing her repeatedly. But it was less the medium than the motive that defined his crime.

Soon after that message, he posted another intended for his ex-partner, Rachelle D'Argent: ''Pay bk u slut.''

A new study of child murders, by Dr Debbie Kirkwood, shows men and women are equally capable of killing children, but their motives for killing are very different. Men are more likely to kill their children in order to take revenge on ex-partners and to make them suffer. Women are more likely to kill because they intend to take their own lives and cannot imagine leaving their children without a mother. Donna Fitchett, who was sentenced to 27 years for the murder of her two sons in 2005, wrote a note to her husband saying she was sorry for his pain. ''I didn't do it because I'm angry with you I just couldn't abandon our beautiful boys ''

It is well-known that separation is a dangerous time for many women, when violence by ex-partners can escalate or be ignited. It is less understood that children can be at serious risk, too, says Kirkwood, the research officer at the Domestic Violence Resource Centre, Victoria. ''People assessing the risk to the mother can assume if there's been no prior violence towards the children, even if there's been violence towards the mother, the children will be OK.''

The study, Just Say Goodbye, draws on court documents to analyse cases where children have died at the hands of parents whose relationship has broken down. The hope is that knowledge of motive and circumstance may lead to better preventive strategies, and improve risk assessment.

Ramazan Acar's motives were spelt out on Facebook, in text messages, and in phone calls that Ms D'Argent received even as she was in the police station with officers who listened in as their colleagues tried to hunt him down. ''I killed her to get back at you. I don't care,'' Acar said. ''Even if I go behind bars, I know that you are suffering.''

As in many - but not all - such cases, violence had been a feature of the couple's eight-year on-again, off-again relationship. At the time of the child's murder, there was an intervention order out against Acar, who pleaded guilty and was sentenced to life imprisonment.

But even more than violence, research shows that a history of obsessive controlling behaviour - a sense of entitlement - can be a pointer to a man's inability to accept the separation. Enduring anger towards his ex-wife who had re-partnered appeared the motive for Robert Farquharson's murder of his three children in 2005. They were killed when he drove them into a dam. Farquharson, convicted at two trials but recently given leave to appeal to the Victorian Court of Appeal, was said by witnesses to be deeply resentful of his former partner. ''I'm going to pay her back big time,'' a witness reported him as saying. ''I'll take away the most important thing to her.''

Arthur Freeman, who threw his four-year-old daughter from the West Gate Bridge in January 2009, was described by his ex-wife as vengeful enough to kill their children to punish her for the separation.

Dean Williamson, who smothered his five year-old son and unsuccessfully tried to kill himself, left a note to his ex-partner that said: ''You're not getting Braddon back, and you're not getting any more money from me again. You are a f---ing useless bitch And you are now going to suffer for the rest of your f---ing useless life.''

Not many children are murdered by their parents - about 27 a year in Australia, according to new data from the National Homicide Monitoring Program. This includes those who died from maltreatment and neglect. It is not known how many of the children died in the context of parental separation. Between 1997 and 2008 there were 110 filicide incidents perpetrated by fathers or stepfathers and 106 by mothers. Many maternal filicides are believed to involve the killing of a baby on the day it is born.

The relatively small number of children murdered in the context of tens of thousands of separations, including many involving violent or controlling men, makes it difficult, Kirkwood admits, to know which children are truly in danger. But in most of the cases in the report the perpetrator had had prior contact with police, courts, mental health services or men's behaviour change programs - missed opportunities for prevention.

''Awareness and understanding are the first step,'' Kirkwood says. ''These are not inexplicable tragedies as is often said.''

While the blame for men's actions is often ascribed to the Family Court system (with the women implicitly held responsible), usually the men have access: Farquharson and Freeman killed their children on access visits; Osborne's children lived with him.

But often obscured is the father's attitude to the mother during and after the relationship, his sense of entitlement to control the family, extreme anger with the mother over separation, and threats to harm the children.

When mothers kill, there is little indication their primary motive is to hurt their partners. Rather, they appear to believe the fathers were uninterested or incapable of looking after the children, the study shows.

Typically, Cathy, who killed one of her children and intended to kill the other two and herself, told police that life was too hard and she could not cope after the breakdown of a second relationship. She said her major dilemma was that, ''If I died, my children would be left on their own. And then they would wonder why their mum had to die. Would they blame themselves?''

For help or information visit Lifeline on 131 114.