Marlin Jackson was described to be very bloody after the incident took place.
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The owner of the dog, North Carolina resident Ronald Mundy Jr., and his dog were involved in the frightening episode moments before take-off.
The flight was traveling from Atlanta to San Diego.
Other passengers on board the plane say the dog was a mixed Lab.
Jackson was seated in a window seat with Mundy and his dog beside him.
"His face was covered in blood, around his eyes, nose cheeks, shirt. When he walked out, he had a cloth over his face and was completely bloody," said passenger Bridget Maddox Peoples.
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Delta Airlines said police were called to the scene and Mundy and his dog were booked on a later flight.
The dog flew in a kennel on the rescheduled flight.
Mundy Jr., a veteran of the Marine Corps, was worried that his dog would be put down following the incident.