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Nondeterministic Graph Searching: From Pathwidth to Treewidth
 In 30th International Symposium on Mathematical Foundations of Computer Science (MFCS), LNCS 3618
, 2005
"... Abstract. We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this gametheoretic approach and graph decom ..."
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Abstract. We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this gametheoretic approach and graph decompositions called qbranched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of qbranched tree decompositions. The equivalence between nondeterministic graph searching and qbranched tree decomposition enables us to design an exact (exponential time) algorithm computing qbranched treewidth for all q ≥ 0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers.
On the Pathwidth of Planar Graphs
, 2006
"... Fomin and Thilikos in [5] conjectured that there is a constant c such that, for every 2connected planar graph G, pw(G) 2pw(G)+c (the same question was asked simutaneously by Coudert, Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph and ac ..."
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Fomin and Thilikos in [5] conjectured that there is a constant c such that, for every 2connected planar graph G, pw(G) 2pw(G)+c (the same question was asked simutaneously by Coudert, Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph and actually is tight by Coudert, Huc and Sereni [4]. In [5], Fomin and Thilikos proved that there is a constant c such that the pathwidth of every 3connected graph G satisfies: pw(G) 6pw(G) + c. In this paper we improve this result by showing that the dual a 3connected planar graph has pathwidth at most 3 times the pathwidth of the primal plus two. We prove also that the question can be answered positively for 4connected planar graphs.
Some Results on Nondeterministic Graph Searching in Trees
"... Pathwidth and treewidth of graphs have been extensively studied for their important structural and algorithmic aspects. Determining these parameters is NPcomplete in general, however it becomes linear time solvable when restricted to some special classes of graphs. In particular, many algorithms ha ..."
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Pathwidth and treewidth of graphs have been extensively studied for their important structural and algorithmic aspects. Determining these parameters is NPcomplete 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 gametheoretical 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 nondeterministic graph searching that provides an unified approach for the pathwidth and the treewidth of a graph. Given q ≥ 0, the qlimited search number, denotes by sq(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, s0(G) corresponds to the pathwidth of a graph G, and s∞(G) corresponds to its treewidth. Fomin et al. proved that computing sq(G) is NPcomplete 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 2approximation of sq(T), for any tree T and any q ≥ 0. This algorithm becomes exact if q ∈ {0, 1}, which proves that the decision problem associated to s1 is polynomial in the class of trees.
Nondeterministic Graph Searching in Trees *
"... Abstract Nondeterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q ≥ 0, the qlimited search number, s q (G), of a graph G is the smallest number of searchers required ..."
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Abstract Nondeterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q ≥ 0, the qlimited search number, s q (G), of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, when the searchers are allowed to know the position of the fugitive at most q times. The search parameter s 0 (G) corresponds to the pathwidth of a graph G, and s ∞ (G) to its treewidth. Determining s q (G) is NPcomplete for any fixed q ≥ 0 in general graphs and s 0 (T ) can be computed in linear time in trees, however the complexity of the problem on trees is open for any q > 0. This paper studies this latter problem. Our main result is the design of a polynomial time algorithm that computes a 2approximation of s q (T ) for any tree T and any q ≥ 0. This algorithm is exact if q ∈ {0, 1}, which proves that the decision problem associated to s 1 is polynomial in the class of trees. To prove this result, we introduce and study a new variant of graph searching that we call restricted. We also prove that the number of queries required to search a tree with two searchers can be computed in linear time. Tight upper bounds on the minimum number of queries for an arbitrary fixed number of searchers is also provided.