Ordered matchings, defined as graphs with linearly ordered vertices, where each vertex is connected to exactly one edge, play a crucial role in the area of ordered graphs and their homomorphisms. Therefore, we consider related problems from the complexity point of view and determine their corresponding computational and parameterized complexities. We show that the subgraph of ordered matchings problem is NP-complete and we prove that the problem of finding ordered homomorphisms between ordered matchings is NP-complete as well, implying NP-completeness of more generic problems. In parameterized complexity setting, we consider a natural choice of parameter - a number of vertices of the image ordered graph. We show that in contrast to the complexity context, finding homomorphisms if the image ordered graph is an ordered matching, this problem parameterized by the number of vertices of the image ordered graph is FPT, which is known to be W[1]-hard for the general problem. We also determine that the problem of core for ordered matchings is solvable in polynomial time which is again in contrast to the NP-completeness of the general problem. We provide several algorithms and generalize some of these problems into ordered graphs with colored edges.

翻译：有序匹配图定义为顶点线性排序的图，其中每个顶点恰好与一条边相连，在有序图及其同态研究领域中具有重要作用。因此，我们从计算复杂性的角度考察相关问题，并确定其对应的计算复杂性及参数化复杂性。我们证明有序匹配子图判定问题是NP完全的，同时验证有序匹配间的有序同态判定问题也是NP完全的，这暗示了更广义问题同样具有NP完全性。在参数化复杂性框架下，我们选取目标有序图的顶点数量作为自然参数。研究表明，与一般情形不同，当目标有序图为有序匹配时，以目标图顶点数为参数的该同态判定问题属于FPT类，而已知一般性问题的该参数化版本为W[1]-难问题。我们还确定有序匹配的核心判定问题可在多项式时间内求解，这再次与一般问题的NP完全性形成对比。我们提出了若干算法，并将部分问题推广至带色边的有序图情形。