Nondeterministic Graph Searching: From Pathwidth to Treewidth

Fomin, Fedor V.; Fraigniaud, Pierre; Nisse, Nicolas

doi:10.1007/11549345_32

Cited by 17 publications

(30 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, this implies that the decision problem related to non-deterministic graph searching is in NP. This also implies that the exponential exact algorithm designed in [10] actually computes s q (G) for any graph G and any q ≥ 0. More interestingly, our result unifies the proof of the monotonicity of invisible graph searching [6] and the proof of the monotonicity of visible graph searching [20].…”

Section: Our Resultsmentioning

confidence: 97%

“…For q ≥ 0, the monotone q-limited search number ms q (G) of a graph G is the smallest number of searchers required to catch any fugitive in G in a monotone way performing at most q queries. The main result of Fomin et al [10] is the following generalization of (1) and (2):…”

Section: Introductionmentioning

confidence: 95%

“…Moreover, Fomin et al [10] prove the NP-completeness of the problem of computing ms q (G), for any q ≥ 0. Using the correspondence between monotone q-limited graph searching and q-branched tree decomposition, they also design an exact exponential algorithm that computes tw q (G) and the corresponding decomposition, for any graph G and any q ≥ 0.…”

Section: Introductionmentioning

confidence: 98%

“…In [10], Fomin et al provide a unified approach to the pathwidth and the treewidth of a graph. For any graph G and any q ≥ 0, they define the notions of q-branched tree decomposition and q-branched treewidth, denoted by tw q (G).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Monotonicity of non-deterministic graph searching

Mazoit¹,

Nisse²

2008

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

In graph searching, a team of searchers are aiming at capturing a fugitive moving in a graph. In the initial variant, called invisible graph searching, the searchers do not know the position of the fugitive until they catch it. In another variant, the searchers permanently know the position of the fugitive, i.e. the fugitive is visible. This latter variant is called visible graph searching. A search strategy that catches any fugitive in such a way that the part of the graph reachable by the fugitive never grows is called monotone. A priori, monotone strategies may require more searchers than general strategies to catch any fugitive. This is however not the case for visible and invisible graph searching. Two important consequences of the monotonicity of visible and invisible graph searching are: (1) the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in NP, and (2) computing optimal search strategies is simplified by taking into account that there exist some that never backtrack.Fomin et al. [F.V. Fomin, P. Fraigniaud, N. Nisse, Nondeterministic graph searching: From pathwidth to treewidth, in: 364-375] introduced an important graph searching variant, called non-deterministic graph searching, that unifies visible and invisible graph searching. In this variant, the fugitive is invisible, and the searchers can query an oracle that permanently knows the current position of the fugitive. The question of the monotonicity of non-deterministic graph searching was however left open.In this paper, we prove that non-deterministic graph searching is monotone. In particular, this result is a unified proof of monotonicity for visible and invisible graph searching. As a consequence, the decision problem corresponding to non-deterministic graph searching belongs to NP. Moreover, the exact algorithms designed by Fomin et al. do compute optimal non-deterministic search strategies.

show abstract

Section: Our Resultsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Monotonicity of non-deterministic graph searching

Mazoit¹,

Nisse²

2008

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…It was proved by Ellis et al [8] that, for any graph G, the node search number of G is equal to the pathwidth of G plus 1. In addition, when the fugitive is visible the cops and robber game becomes the equivalent of the treewidth, and Fomin et al [9] filled the gap between treewidth and pathwidth introducing in the cops and robber game a parameter controlling the number of times the fugitive is visible.…”

mentioning

confidence: 99%

A Distributed Algorithm for Computing the Node Search Number in Trees

2011

View full text Add to dashboard Cite

International audienceWe present a distributed algorithm to compute the node search number in trees. This algorithm extends the centralized algorithm proposed by Ellis \emph{et al.}~[EST94]. It can be executed in an asynchronous environment, requires an overall computation time of $O(n\log{n})$, and $n$ messages of $\log_3{n}+4$ bits each. The main contribution of this work lies in the data structure proposed to design our algorithm, called \emph{hierarchical decomposition}. This simple and flexible data structure is used for four operations: updating the node search number after addition or deletion of any tree-edges in a distributed fashion; computing it in a tree whose edges are added sequentially and in any order; computing other graph invariants such as the process number and the edge search number, by changing only initialization rules; extending our algorithms for trees and forests of unknown size (using messages of up to $2\log_3{n}+5$ bits)

show abstract

Connected search for a lazy robber

Adler

Paul

Thilikos

2021

Journal of Graph Theory

View full text Add to dashboard Cite

The node-search game against a lazy (or, respectively, agile) invisible robber has been introduced as a searchgame analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above games, we additionally demand that, at each moment of the search, the clean territories are connected.1 Interestingly, the motivating story of one of the foundational articles on graph searching, authored by Torrence Parsons [46] in 1976, was inspired by an earlier article of Breisch in Southwestern Cavers Journal [13] proposing a "speleotopological" approach for the problem of finding an explorer lost in a system of dark caves. It is worth to stress that that setting neglected the natural connectivity requirement.

show abstract

Nondeterministic Graph Searching: From Pathwidth to Treewidth

Cited by 17 publications

References 20 publications

Monotonicity of non-deterministic graph searching

Monotonicity of non-deterministic graph searching

A Distributed Algorithm for Computing the Node Search Number in Trees

Connected search for a lazy robber

Contact Info

Product

Resources

About