Finding a simple path of even length between two designated vertices in a directed graph is a fundamental NP-complete problem known as the EvenPath problem. Nedev proved in 1999, that for directed planar graphs, the problem can be solved in polynomial time. More than two decades since then, we make the first progress in extending the tractable classes of graphs for this problem. We give a polynomial time algorithm to solve the EvenPath problem for classes of H-minor-free directed graphs,1 where H is a single-crossing graph. We make two new technical contributions along the way, that might be of independent interest. The first, and perhaps our main, contribution is the construction of small, planar, parity-mimicking networks. These are graphs that mimic parities of all possible paths between a designated set of terminals of the original graph. Finding vertex disjoint paths between given source-destination pairs of vertices is another fundamental problem, known to be NP-complete in directed graphs, though known to be tractable in planar directed graphs. We encounter a natural variant of this problem, that of finding disjoint paths between given pairs of vertices, but with constraints on parity of the total length of paths. The other significant contribution of our paper is to give a polynomial time algorithm for the 3-disjoint paths with total parity problem, in directed planar graphs with some restrictions (and also in directed graphs of bounded treewidth).

翻译：在有向图中寻找两个指定顶点间长度为偶数的简单路径，是一个被称为EvenPath问题的基本NP完全问题。Nedev于1999年证明，对于有向平面图，该问题可在多项式时间内求解。此后二十余年间，我们在扩展该问题的可处理图类方面首次取得进展。针对H次小有向图类（其中H为单交叉图），我们给出了求解EvenPath问题的多项式时间算法。在此过程中，我们取得了两项可能具有独立价值的新技术贡献。第一个（或许也是我们的主要）贡献是构建了小型平面奇偶模拟网络。这类网络能够模拟原图中指定终端集之间所有可能路径的奇偶性。在给定源-目标顶点对之间寻找顶点不相交路径是另一个基本问题，已知在有向图中是NP完全的，但在有向平面图中是可处理的。我们遇到了该问题的一个自然变体：在给定顶点对间寻找不相交路径，但要求路径总长度满足特定奇偶约束。本文的另一重要贡献是，在具有特定限制条件的有向平面图（以及有界树宽有向图）中，为带总奇偶约束的3-不相交路径问题提供了多项式时间算法。