For automatic and recursive graphs, we investigate the following problems:
(A) existence of a Hamiltonian path and existence of an infinite path in a tree
(B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree
(C) existence of an infinite clique and an infinite version of set cover
(A) are equally complex for automatic and for recursive graphs (Σ11-complete),
(B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy),
(C) are much simpler for automatic than for recursive graphs (decidable and Σ11-complete, resp.).
"Some natural decision problems in automatic graphs." J. Symbolic Logic 75 (2) 678 - 710, June 2010. https://doi.org/10.2178/jsl/1268917499