No, there are three 'dog' characters in the movie Labyrinth. Merlin/Ambrosious, and Sir Didymous.
I believe the movie you are thinking of is The Never Ending Story, it has a Luck Dragon named Falcor that resembles a giant flying dog.
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In the movie the dog breed was a St Bernard
In the books Fang is a boarhound. But the dog playing Fang in the movies a Neapolitan Mastiff.
its a Labrador Retriever
William Skidmore was played by Tiko Martinez in the movie Alpha Dog.
I saw the movie and I just realise that the dog was a great dane, one of my favourite!