The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label.

In this paper, we present a solution to finding an s–t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s–t path in a ${\mathbb{Z}}_3$-labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s–t paths in a group-labeled graph or not, and finding s–t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s–t paths, which is the main technical contribution of this paper.

路径和循环长度的奇偶性是图论和理论计算机科学中一个经典且经过充分研究的主题。奇偶性约束可以扩展到组标记图中的标记约束，该组标记图是有向图，每个圆弧都由组的元素标记。最近，已经研究了带有组标记的图中的路径和循环，例如打包非零路径和循环，其中“非零”表示标识元素是唯一的禁止标签。

在本文中，我们提出了一种在带有组标签的图中禁止使用两个标签的s – t路径的解决方案。这也导致了一个基本的解决方案，即找到一个零s – t路径。${\mathbb{ž}}_3$标记的图，这是找到零路径的第一个重要案例。实际上，这种情况在无向图中推广了2不相交路径问题，这也促使我们考虑该设置。更准确地说，我们提供了多项式时间算法，用于测试在带有组标记的图中是否最多存在s - t条路径的两个可能的标签，并找到存在至少三个不同标签的s - t条路径。该算法基于组标记图具有两个可能的s – t路径标记的必要和充分条件，这是本文的主要技术贡献。