First-order query evaluation on structures of bounded degree

We consider the enumeration problem of first-order queries over structures of bounded degree. Durand and Grandjean have shown that this problem is in CONSTANT-DELAY (lin). An enumeration problem belongs to CONSTANT-DELAY(lin) if for an input of size n it can be solved by (i) an O(n) precomputation phase building an index structure, followed by (ii) a phase enumerating the answers with no repetition and a constant delay between two consecutive outputs. In this article we give a different proof of this result based on Gaifman's locality theorem for first-order logic. Moreover, the constants we obtain yield a total evaluation time that is triply exponential in the size of the input formula, matching the complexity of the best known evaluation algorithms.