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We use automata-theoretic approach to analyze properties of Fibonacci words. The directed acyclic subword graph (dawg) is a useful deterministic automaton accepting all suffixes of the word. We show that dawg's of Fibonacci words have particularly simple structure. The simple structure of paths in these graphs gives simplified alternative proofs and new interpretation of several known properties of Fibonacci words. The structure of lengths of paths in the compacted subword graph corresponds to a number-theoretic characterization of occurrences of subwords in terms of Zeckendorff Fibonacci number system. Using the structural properties of dawg's it can be easily shown that for a string w we can check if w is a subword of a Fibonacci word in time O(|w|) and O(1) space. Compact dawg's of Fibonacci words show a very regular structure of their suffix trees and show how the suffix tree for the Fibonacci word grows (extending the leaves in a very simple way) into the suffix tree for the next Fibonacci word.