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New update since first reported:  — The Houston family says, “Bobbi Kristina is fighting for her life and is surrounded by immediate family. As her father already stated, we are asking you to honor our request for privacy during this difficult time.”

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Things are touch and go for the daughter of the late Whitney Houston, Bobbi Kristina, but according to doctors just hours ago, she is showing some significant improvement.

TMZ reports that The Houston family is saying that, “Bobbi Kristina is fighting for her life and is surrounded by immediate family. As her father already stated, we are asking you to honor our request for privacy during this difficult time.”

Sources connected with the family tell us … doctors performed tests Monday morning and determined the swelling on her brain has gone down. In addition, she has been able to “move her eyes” … all important signs.

Perhaps most important … we’re told the oxygen levels in her brain have increased significantly — 4 times better than Sunday.

But the family has been told … because Bobbi Kristina was deprived of oxygen for anywhere between 2 to 5 minutes while she was submerged in a tub, doctors fear neurological damage.

WRRR reported several days ago that Bobbi Kristina Brown, the daughter of the late singer Whitney Houston, was found unresponsive in a bathtub in her Georgia home last Saturday, police said. Brown’s husband, Nick Gordon, and a friend found her in the tub in the morning and started CPR, the Roswell Police Department said in a statement.