We make progress on a generalization of the road (colouring) problem. The road problem was posed by Adler-Goodwyn-Weiss and solved by Trahtman. The generalization was posed, and solved in certain special cases, by Ashley-Marcus-Tuncel. We resolve two new families of cases, of which one generalizes the road problem and follows Trahtman's solution, and the other generalizes a result of Ashley-Marcus-Tuncel with a proof quite different from theirs. Along the way, we prove a universal property for the fiber product of certain graph homomorphisms, which may be of independent interest. We provide polynomial-time algorithms for relevant constructions and decision problems.

翻译：道路问题由Adler-Goodwyn-Weiss-Weiss提出,由Trahtman解决。一般化是由Ashley-Marcus-Tuncel提出,在某些特殊情况下由Ashley-Marcus-Tuncel提出。我们解决了两个新的案例,其中一个概括了道路问题,并遵循Trahtman的解决方案,另一个概括了Ashley-Marcus-Tuncel的结果,其证据与它们大相径庭。与此同时,我们证明某些图形同质体纤维产品具有普遍特性,可能具有独立的兴趣。我们为相关的构建和决定问题提供了多时算法。