The second most important of a photograph is composition, the most important part is light and I will write on that seperately.
Composition makes or breaks the photograph, it can engage the viewer or make the photograph invisible because it fails to draw the viewers attention.
When photographing dogs, the most important element is the eyes, the need to sharp and in focus, another important aspect is expression. The most expressive part of a dog is the head as you have the eyes, ears and head tilt all of which combine for eye catching expressions that will draw the viewer but it is still the composition that needs to be correct.
That is a very interesting statement because what is correct composition? It is whatever you think works because you are taking the photograph and your vision is unique to you.
You do have an obligation to capture images the dogs owner will love also so you have to take many pictures but your composition is still the key, if the image is sharp and the owner can see the dogs eyes, they will largely be happy and not pay much attention to the composition if the composition is correct.
If the composition is not correct the owner will notice because something in the picture has drawn there attention away from their dog.
Cluttered backgrounds can be a big source of distraction because we are so focussed n the dog through the viewfinder we stop looking to see what is around the dog. If the back ground is simple and plain, it is easier to blur by adjusting your depth of field. if it is cluttered, it is much harder to blur to make is more insignificant.
Where you place the dog in the frame can make or break your photograph so have a think where you want the dog to be, either move your focus point or focus and recompose to achieve your desired effect and try lots os different positions in the frame as you can create very different pictures by placing the dog in different parts of the frame.
Get close to the dog, be bold, be creative and don't be afraid to make a mistake, it is free to delete an image!