Help making some custom string

Okay so ill explain this as best i can, i want a string that allows the widget to show if its
1. the category page of post category “dogs”
2. a child category of post category “dogs” (ie category “poodles”, “pugs”, “staffy” which has parent category “dogs”
3. a single post within either of the above categories (ie in “dogs”, or in a child category of “poodles”, “pugs”, “staffy”)

Ive played around and i just cant seem to figure it out, im sure it would be breeze for you smart people.

Thanks