The women were defending a toddler.
Kati Mather was left in critical condition with more than 100 bites across her body, and a Good Samaritan was also injured.
"This afternoon, the city filed an application in provincial court for an order to have the animal destroyed," said Ted Townsend, a spokesperson for the City of Richmond.
"Our application is based on the severity of the attack and the threat to public safety."
Townsend said the injunction allows the city to retain custody of the dog, which has been kept in isolation at the Richmond animal shelter since the attack.
Yogi's owner, Lucas MacNeil, has defended the dog and says it shouldn't be destroyed.
He has been posting photos and videos of Yogi on social media as part of a social media campaign that portrays him as a sweet, lovable dog.
MacNeil is also the boyfriend of Mather's twin sister, Jessi. The two women were playing in the park with Jessi's three-year-old son when the dog attacked, according to the young boy's father.
Officers found Kati Mather on the ground, covered in blood and trying to fend off the 35-kilogram dog.
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