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Rapport (Rapport De Recherche) Année : 2007

Some Results on Non-deterministic Graph Searching in Trees

Résumé

Pathwidth and treewidth of graphs have been extensively studied for their important structural and algorithmic aspects. Determining these parameters is NP-complete in general, however it becomes linear time solvable when restricted to some special classes of graphs. In particular, many algorithms have been proposed to compute efficiently the pathwidth of trees. Skodinis (2000) proposes a linear time algorithm for this task. Pathwidth and treewidth have also been studied for their nice game-theoretical interpretation, namely graph searching games. Roughly speaking, graph searching problems look for the smallest number of searchers that are sufficient to capture a fugitive in a graph. Fomin et al. (2005) define the non-deterministic graph searching that provides an unified approach for the pathwidth and the treewidth of a graph. Given q>=0, the q-limited search number, denotes by s_q(G), of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, such that the searchers are allowed to know the position of the fugitive at most q times. Roughly, s_0(G) corresponds to the pathwidth of a graph G, and s_\infty(G) corresponds to its treewidth. Fomin et al. proved that computing s_q(G) is NP-complete in general, and left open the complexity of the problem restricted to the class of trees. This paper studies this latter problem. On one hand, we give tight upper bounds on the number of queries required to search a tree when the number of searchers is fixed. We also prove that this number can be computed in linear time when two searchers are used. On the other hand, our main result consists in the design of a simple polynomial time algorithm that computes a 2-approximation of s_q(T), for any tree T and any q>=0. This algorithm becomes exact if q=0 or 1, which proves that the decision problem associated to s_1 is polynomial in the class of trees.
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Dates et versions

inria-00174965 , version 1 (26-09-2007)
inria-00174965 , version 2 (26-09-2007)
inria-00174965 , version 3 (11-10-2013)

Identifiants

  • HAL Id : inria-00174965 , version 2

Citer

Omid Amini, David Coudert, Nicolas Nisse. Some Results on Non-deterministic Graph Searching in Trees. [Research Report] 2007, pp.18. ⟨inria-00174965v2⟩
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