We exploit a decomposition of graph traversals to give a novel characterization of depth-first and breadth-first traversals as universal constructions. Specifically, we introduce functors from two different categories of edge-ordered directed graphs into two different categories of transitively closed edge-ordered graphs; one defines the lexicographic depth-first traversal and the other the lexicographic breadth-first traversal. We show that each functor factors as a composition of universal constructions, and that the usual presentation of traversals as linear orders on vertices can be recovered with the addition of an inclusion functor. Finally, we raise the question of to what extent we can recover search algorithms from the categorical description of the traversal they compute.

中文翻译：

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图遍历为通用构造

我们利用图遍历的分解来给出深度优先遍历和广度优先遍历的新颖特征，作为通用构造。具体来说，我们将函子从两个不同类别的有序有向图引入到两个不同类别的传递闭合式有序图中；一个定义字典式深度优先遍历，另一个定义字典式宽度优先遍历。我们表明，每个函子因子都是通用构造的组成，并且可以通过添加包含函子来恢复遍历作为顶点在顶点上的线性顺序的通常形式。最后，我们提出一个问题，即我们可以在多大程度上从它们计算的遍历的分类描述中恢复搜索算法。