# Graph-Minor Theorem for Directed Graphs?

Suppose that $\vec{G}$ is a directed graph and that $G$ is the undirected graph obtained from $\vec{G}$ by forgetting the direction on each edge. Define $\vec{H}$ to be a minor of $\vec{G}$ if $H$ is a minor of $G$ as undirected graphs and direction on the edges of $\vec{H}$ are the same as the corresponding edges in $\vec{G}$.

Does the Robertson-Seymour Theorem hold for directed graphs (where the above definition of minor is used and our graphs are allowed to have loops and multiple edges)?

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No. Digraphs are not well-quasi ordered under minor containment. However a subclass of digraph, namely semi-complete tournament (a fully connected DAG), are WQO under minor containment. Source: recent (2011-2012) paper by Seymour.

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10.5 Let $G_i$ ($i = 1,2,\ldots$) be a countable sequence of directed graphs. Then there exist $j > i \geq 1$ such that $G_i$ is isomorphic to a minor of $G_j$.