Connected graph searching

In the graph searching game the opponents are a set of searchers and a fugitive in a graph. The searchers try to capture the fugitive by applying some sequence of moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive. In this paper, we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean (i.e. non-accessible to the fugitive) remain connected. We give evidence that many of the standard mathematical tools used so far in classic graph searching fail under the connectivity requirement. We also settle the question on “the price of connectivity”, that is, how many searchers more are required for searching a graph when the connectivity demand is imposed. We make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees. In particular, for an n-vertex graph the ratio between the connected searching number and the non-connected one is O(log n) while for trees this ratio is always at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for all graphs. Our combinatorial results imply a complete characterization of connected graph searching on trees. It is based on a forbidden-graph characterization of the connected search ∗Results of this paper have appeared in the proceedings of the 14th ACM Symposium on Parallel Algorithm and Architecture (SPAA 2002), the 29th Workshop on Graph Theoretic Concepts in Computer Science (WG 2003), the 13th SIAM Conference on Discrete Mathematics, (DM 2006), and the 7th Latin American Theoretical Informatics Symposium (LATIN 2006). †Correspondance author: Nicolas Nisse. ‡Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Spain. http://wwwma4.upc.edu/ ̃lali §School of Information Technology and Engineering, University of Ottawa, Canada. http://www.site.uottawa.ca/ ̃flocchin/ Supported in part by NSERC Discovery Grant. ¶Department of Informatics, University of Bergen, Norway. http://www.ii.uib.no/ ̃fomin ‖CNRS and Université Paris Diderot, France. http://www.liafa.jussieu.fr/ ̃pierref/ Supported by the ANR projects ALADDIN and PROSE, and by the INRIA project GANG. ∗∗MASCOTTE, INRIA, I3S (CNRS/Université Nice Sophia Antipolis), France. http://wwwsop.inria.fr/members/Nicolas.Nisse/ Supported by the ANR project AGAPE. ††School of Computer Science, Carleton University, Canada. http://www.scs.carleton.ca/ ̃santoro Supported in part by NSERC Discovery Grant. ‡‡Department of Mathematics, National and Kapodistrian University of Athens, Greece. http://www.thilikos.info Supported by the project “Kapodistrias” (AΠ 02839/28.07.2008) of the National and Kapodistrian University of Athens (project code: 70/4/8757).