Odysseus got to see his old dog after 20 years, the dog was the only person that remembered him. He sees the dog has been treated very poorly and thinks he could have been a great dog if he had not been. Later on, the dog passes away after holding on for 20 years and getting to see his master one last time.
This part of the story connects with the rest because it gives an insight into how they lived. The dog being treated this way is not only reflective of the dog but the place as a whole. It shows that the whole area is falling apart. It also shows us Odysseus is not remebered.
I marvel that they leave this find dog here on a dung pile, he would have been a wonderful dog of they let him.