Cops-and-robbers: remarks and problems Michel Boyer a,1 , Sif El Harti a , Amal El Ouarari a , Robert Ganian b , Geˇ na Hahn a,1,1 , Carsten Moldenauer c , Ignaz Rutter d,1 , Benoit Th´ eriault a,2 , Martin Vatshelle 1,e a D´ epartement d’informatique et de recherche op´ erationnelle, Universit´ e de Montr´ eal, C.P. 6126 succursale Centre-ville, Montr´ eal, QC, Canada, H3C 3J7 b Faculty of Informatics, Masaryk University, Botanick´a 68a, 60200 Brno Czech Republic c Humboldt-Universit¨at zu Berlin, Institut f¨ ur Informatik, D-10099 Berlin, Germany d Karlsruhe Institute of Technology (KIT), Institute of Theoretical Informatics, D-76128 Karlsruhe, Germany e University of Bergen, Department of Informatics, pb. 7803, 5020 Bergen, Norway Abstract We explore cops-and-robbers games in several directions, giving partial re- sults in each and refuting two reasonable conjectures. We close with some open problems. Keywords: cops-and-robbers, graph, directed graph, tournament, reflexive graph, graph searching, optimal game, constrained game 1. Introduction and basics The purpose of this short paper is to collect partial results from several M.Sc. theses, to indicate some unexplored directions of research and to refute several conjectures that seemed reasonable at the time they were made. Further, we suggest a new - and we hope useful - generalization and prove a few results about it. The common beginning is the original cop-and-robber Email addresses: boyer@iro.umontreal.ca (Michel Boyer), 99352@mail.muni.cz (Robert Ganian), hahn@iro.umontreal.ca (Geˇ na Hahn), moldenha@informatik.hu-berlin.de (Carsten Moldenauer), rutter@kit.edu (Ignaz Rutter), Martin.Vatshelle@ii.uib.no (Martin Vatshelle) 1 The author gratefully acknowledges the support of the Natural Science and Engineer- ing Research Council of Canada. 2 The author gratefully acknowledges the support of NSERC’s Undergraduate Student Research Award program. Preprint submitted to Elsevier October 7, 2011