The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for $H$-free graphs. If $k$ is fixed, we obtain the $k$-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every $k \geq 1$. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of $k$-Disjoint Connected Subgraphs for $H$-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for $H$-free graphs as for Disjoint Paths.

中文翻译：

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无H图的不相交路径和连通子图

众所周知的“不相交路径”问题是确定一个图是否包含k个成对的不相交路径，每个路径都连接来自k个不同对的集合中的不同终端对。除两种情况外，我们确定无$ H $的图的不相交路径问题的复杂性。如果$ k $是固定的，我们将获得$ k $ -Disjoint Paths问题，这对于每张$ k \ geq 1 $在所有图的类中都是多项式时间可解的。如果我们需要连接端子组而不是端子对中的顶点，则后者不再成立。对于无$ H $的图，我们将$ k $-不相交的连通子图的复杂性完全分类，并且为无$ H $的图给出不相交的连通子图的几乎完全相同的分类。