Question about Lawn - Trying to grow grass

First off let me explain what I have:

My wife and I have been in our home a year now. The previous owners had 2 labs that they kept in a pen in the backyard. Around the back porch they built a small fence to keep the dogs from coming up on the porch. You can see the path(s) the dogs took up and down the fence line and up around the back of the house as there is no grass there anymore. I know this house used to have a lot of grass in the backyard because it's in the same neighborhood that my grandparents lived in.

There is one big oak tree in the backyard that provides a good bit of shade. My neighbors also have oak and pine trees that provide shade.

One of my friends told me that I should bring in a mixture of top soil / sand and spread it out in the bare areas and then put down some sod. From what I can tell, the grass is mostly centipede.

What advice would you give or would you need a picture or two to be able to give better advice?