Graph Traversals as Universal Constructions

Authors Siddharth Bhaskar , Robin Kaarsgaard

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Siddharth Bhaskar
  • Department of Computer Science, University of Copenhagen, Denmark
Robin Kaarsgaard
  • School of Informatics, University of Edinburgh, UK


We would like to thank Steve Lindell and Scott Weinstein for inspiring us to study traversals in terms of their predecessor functions, as well as suggesting the characterization of breadth-first traversals in Corollary 27. We also thank Jade Master for discussions relating to this work. We are indebted to the anonymous reviewers for their thorough and detailed comments.

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Siddharth Bhaskar and Robin Kaarsgaard. Graph Traversals as Universal Constructions. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We exploit a decomposition of graph traversals to give a novel characterization of depth-first and breadth-first traversals by means of 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Theory of computation → Models of computation
  • graph traversals
  • adjunctions
  • universal constructions
  • category theory


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