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The longest route in a graph that has a specific direction and does not contain cycles is called a "directed acyclic path" or "DAG". It can also be referred to as a "topological order" of the graph. This type of path is important in many applications, such as scheduling and task ordering, because it represents a sequence of events that must occur without any circular dependencies or conflicts. The longest DAG can be found using algorithms such as topological sorting, which orders the vertices of the graph in a way that respects the direction of the edges and ensures that no cycles are present. The length of the longest DAG is equal to the length of the longest path from a source vertex to a sink vertex in the graph.