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：我们在道路（着色）问题的一个推广上取得了进展。道路问题由Adler-Goodwyn-Weiss提出，并由Trahtman解决。该推广由Ashley-Marcus-Tuncel提出，并在某些特殊情况下得到了解决。我们解决了两个新的情形家族，其中一种推广了道路问题并沿用了Trahtman的解法，另一种推广了Ashley-Marcus-Tuncel的一个结果，但其证明方法与他们的截然不同。在此过程中，我们证明了某些图同态的纤维积具有普适性质，该性质可能具有独立的研究价值。我们为相关构造和判定问题提供了多项式时间算法。