Cops and robbers on oriented graphs Devvrit Khatri 1 , Natasha Komarov 2 , Aaron Krim-Yee 3 , Nithish Kumar 4 , Ben Seamone 5,6 , Virg ´ elot Virgile 7 , and AnQi Xu 5 1 Department of Computer Science and Engineering, Birla Institute of Technology and Science, Pilani, India 2 Department of Mathematics, Computer Science, and Statistics, St. Lawrence University, Canton, NY, USA 3 Department of Bioengineering, McGill University, Montreal, QC, Canada 4 Department of Computer Science and Engineering, National Institute of Technology – Tiruchirapalli, India 5 Mathematics Department, Dawson College, Montreal, QC, Canada 6 D´ epartement d’informatique et de recherche op´ erationnelle, Universit´ e de Montr´ eal, Montreal, QC, Canada 7 D´ epartement de math´ ematiques et de statistique, Universit´ e de Montr´ eal, Montreal, QC, Canada January 13, 2020 Abstract We consider the well-studied cops and robbers game in the context of oriented graphs, which has received surprisingly little attention to date. We examine the relationship between the cop numbers of an oriented graph and its underlying undirected graph, giving a surprising result that there exists at least one graph G for which every strongly connected orientation of G has cop number strictly less than that of G. We also refute a conjecture on the structure of cop-win digraphs, study orientations of outerplanar graphs, and study the cop number of line digraphs. Finally, we consider some the aspects of optimal play, in particular the capture time of cop-win digraphs and properties of the relative positions of the cop(s) and robber. 1 arXiv:1811.06155v1 [math.CO] 15 Nov 2018