Connected Graph Searching - Archive ouverte HAL Access content directly
Journal Articles Information and Computation Year : 2012

Connected Graph Searching

(1) , (2) , (3) , (4) , (5) , (6) , (7, 8)
1
2
3
4
5
6
7
8

Abstract

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 number. We prove that the connected search game is monotone for trees, i.e. restricting search strategies to only those where the clean territories increase monotonically does not require more searchers. A consequence of our results is that the connected search number can be computed in polynomial time on trees, moreover, we show how to make this algorithm distributed. Finally, we reveal connections of this parameter to other invariants on trees such as the Horton-Stralher number.
Fichier principal
Vignette du fichier
journalFinal.pdf (573.2 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00741948 , version 1 (15-10-2012)

Identifiers

  • HAL Id : hal-00741948 , version 1

Cite

Lali Barrière, Paola Flocchini, Fedor V. Fomin, Pierre Fraigniaud, Nicolas Nisse, et al.. Connected Graph Searching. Information and Computation, 2012, 219, pp.1-16. ⟨hal-00741948⟩
632 View
161 Download

Share

Gmail Facebook Twitter LinkedIn More