A man who allegedly set a woman on fire in San Francisco on Sunday had had a “dating relationship” with the victim, police said.

San Francisco police were searching for the suspect, Dexter Oliver, 22, who faces charges of attempted murder and arson in the attack that left the woman with “life-threatening injuries,” San Francisco police Officer Carlos Manfredi said.

Oliver is 5 feet 10 and weighs 155 pounds. He wears a hair cut with a fish pattern on one side and a lightning bolt on the other. Manfredi said Oliver was wearing a red and white hoodie, pajama pants with a multicolored shamrock design and grey and red Nike shoes.

Police responded to reports of a woman screaming at 12:17 p.m. in the city’s Bayview District. “He approached her and threw ignitable liquid on top of her, causing her to suffer serious burns,” Manfredi said.

The woman was taken to St. Francis Hospital's burn center, where she was in critical condition. It was not clear what prompted the attack, Manfredi said.

San Francisco police have not released the victim's name, but she was identified by relatives as 25-year-old Starr Lamare, according to KGO-TV in San Francisco.

The victim's sister, Precious Craig, told the news station that Lamare and Oliver had a heated argument over whether to leave laundry at a laundromat when he allegedly rushed out to get gasoline and then doused her with it. Witnesses interviewed by KGO-TV said the victim was set on fire, then managed to take off her burning sweater as she ran a few blocks down the street toward her home. Emergency crews responding to 911 calls picked her up and rushed her to the hospital.

"Most of her face is burnt," Craig told KGO-TV. "Just a little bit of hair is still there, but most of her face is burned."

Lamare is the mother of three children and had only been dating the suspect for about six months, the family told KGO-TV.

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--Teresa Watanabe