Connected Graph Searching 1 2

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 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). Paola Flocchini and Nicola Santoro are supported in part by NSERC Discovery Grant; Pierre Fraigniaud is supported by the ANR projects ALADDIN and PROSE, and by the INRIA project GANG; Nicolas Nisse is supported by the ANR project AGAPE; Dimitrios M. Thilikos is supported by the project “Kapodistrias” (AΠ 02839/28.07.2008) of the National and Kapodistrian University of Athens (project code: 70/4/8757). Preprint submitted to Elsevier August 22, 2012 ha l-0 07 41 94 8, v er si on 1 15 O ct 2 01 2 Author manuscript, published in "Information and Computation 219 (2012) 1-16"