On the Tree-Width of Planar Graphs

Dieng, Youssou; Gavoille, Cyril

doi:10.1016/j.endm.2009.07.099

Cited by 5 publications

(4 citation statements)

References 6 publications

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“…We observe that a separator of size O δ (log(n)) that is not in-class can be easily obtained by combining two existing results. Chepoi et al [25,Proposition 13] bounded the treelength of δ-hyperbolic graphs, and Dieng and Gavoille [40] (see also [39]) bounded the treewidth of a planar graph in terms of its treelength, which gives the following bound on the treewidth of planar δ-hyperbolic graphs: 25] and [40]). For any δ ⩾ 0, the treewidth of any n-vertex planar δ-hyperbolic graph is O(δ log n).…”

Section: Main Contribution: a Novel Separator Theoremmentioning

confidence: 99%

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Kisfaludi-Bak

Nederlof

Węgrzycki

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity.Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ ⩾ 0, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph.An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G − Z induces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is poly(δ) • log n.As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running inWe also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no n o(δ) -time algorithm on planar δ-hyperbolic graphs, unless ETH fails.

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Section: Main Contribution: a Novel Separator Theoremmentioning

confidence: 99%

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Kisfaludi-Bak

Nederlof

Węgrzycki

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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“…For instance, a clique on n vertices has tree-length 1 and tree-width n − 1, whereas a cycle on 3n vertices has tree-width 2 and tree-length n. However, if one involves also the size (G) of a largest isometric cycle in G, then a relation is possible: tl(G) ≤ (G)/2 (tw(G) − 1) [21] (see also [3] and [25] for similar but slightly weaker bounds: [25]). Furthermore, the tree-width of a planar graph G is bounded by O(tl(G)) [24].…”

Section: Slimness and Tree-lengthmentioning

confidence: 99%

Slimness of graphs

Dragan,

Mohammed

2017

Preprint

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Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph G = (V, E), a geodesic triangle (x, y, z) with x, y, z ∈ V is the union P (x, y) ∪ P (x, z) ∪ P (y, z) of three shortest paths connecting these vertices. A geodesic triangle (x, y, z) is called δ-slim if for any vertex u ∈ V on any side P (x, y) the distance from u to P (x, z) ∪ P (y, z) is at most δ, i.e. each path is contained in the union of the δ-neighborhoods of two others. A graph G is called δ-slim, if all geodesic triangles in G are δ-slim. The smallest value δ for which G is δ-slim is called the slimness of G. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter ∆(G) of a layering partition of G, (2) graphs with tree-length λ, (3) graphs with tree-breadth ρ, (4) k-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1.

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“…Note that such graphs have a large genus, i.e., they are in a sense arbitrarily far from planar graphs. In contrast, it holds that tw(G) < 12 • tl(G) for planar graphs [17]. Consequently, it is quite natural to ask whether a treewidth arbitrarily larger than the treelength requires a large genus.…”

Section: State Of the Artmentioning

confidence: 99%

To Approximate Treewidth, Use Treelength!

Coudert

Ducoffe

Nisse

2016

SIAM J. Discrete Math.

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International audienceTree-likeness parameters have proven their utility in the design of efficient algorithms on graphs. In this paper, we relate the structural tree-likeness of graphs with their metric tree-likeness. To this end, we establish new upper-bounds on the diameter of minimal separators in graphs. We prove that in any graph G, the diameter of any minimal separator S in G is at most ⌊l(G)/2⌋ · (|S| − 1), with l(G) the length of a longest isometric cycle in G. Our result relies on algebraic methods and on the cycle basis of graphs. We improve our bound for the graphs admitting a distance preserving elimination ordering, for which we prove that any minimal separator S has diameter at most 2 · (|S| − 1). We use our results to prove that the treelength tl(G) of any graph G is at most ⌊l(G)/2⌋ times its treewidth tw(G). In addition, we prove that, for any graph G that excludes an apex graph H as a minor, tw(G) ≤ c_H · tl(G) for some constant c_H only depending on H. We refine this constant when G has bounded genus. Altogether, we obtain a simple O(l(G))-approximation algorithm for computing the treewidth of n-node apex-minor-free graphs in O(n^2)-time

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On the Tree-Width of Planar Graphs

Cited by 5 publications

References 6 publications

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Slimness of graphs

To Approximate Treewidth, Use Treelength!

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