Forbidden graphs for tree-depth

Dvořák, Zdenk; Giannopoulou, Archontia C.; Thilikos, Dimitrios M.

doi:10.1016/j.ejc.2011.09.014

Cited by 33 publications

(34 citation statements)

References 11 publications

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“…Lagergren [61] showed that the number of edges in every obstruction to a graph of treewidth k is at most double exponential in O(k 5 ). Dvořák et al [40] provide similar bound on obstructions to graphs of tree-depth at most k. Dinneen and Xiong have shown that the number of vertices in connected obstruction for graphs with vertex cover at most k is at most 2k + 1 [37]. Obstructions for graphs with feedback vertex set of size at most k is discussed in the work of Dinneen et al [36].…”

mentioning

confidence: 69%

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Fomin

Lokshtanov²,

Misra

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

128

174

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Let F be a finite set of graphs. In the F-Deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth η-Deletion.In this paper we obtain a number of generic algorithmic results about F-Deletion, when F contains at least one planar graph. The highlights of our work are• A constant factor approximation algorithm for the optimization version of F-Deletion;• A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2 O(k) n), for the parameterized version of F-Deletion where all graphs in F are connected;• A polynomial kernel for parameterized F-Deletion.These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results -constant factor approximation, polynomial kernelization and FPT algorithms -are stringed together by a common theme of polynomial time preprocessing.

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mentioning

confidence: 69%

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Fomin

Lokshtanov²,

Misra

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

128

174

View full text Add to dashboard Cite

show abstract

“…However, a complete classification of minimal obstructions for treedepth ≤ k remains elusive even for small values of k (less than 10). Moreover, Dvořák et al [14] showed that the number of minimal obstructions grows enormously fast (at least doubly exponentially) as a function of k [14]. The situation is similar for other width measures like treewidth.…”

Section: Introductionmentioning

confidence: 73%

“…for specific minor-monotone graph invariants is an active topic of research in graph theory (see [1,11]). When it comes to treedepth, minimal obstructions have been studied by two sets of authors [4,5,14,16]. However, a complete classification of minimal obstructions for treedepth ≤ k remains elusive even for small values of k (less than 10).…”

Section: Introductionmentioning

confidence: 99%

A Polynomial Excluded-Minor Approximation of Treedepth

Kawarabayashi

Rossman

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Treedepth is a well-studied graph invariant in the family of "width measures" that includes treewidth and pathwidth. Understanding these invariants in terms of excluded minors has been an active area of research. The recent Grid Minor Theorem of Chekuri and Chuzhoy [12] establishes that treewidth is polynomially approximated by the largest k × k grid minor. In this paper, we give a similar polynomial excluded-minor approximation for treedepth in terms of three basic obstructions: grids, tree, and paths. Specifically, we show that there is a constant c such that every graph of treedepth ≥ k c contains one of the following minors (each of treedepth ≥ k):• the complete binary tree of height k,• the path of order 2 k .Let us point out that we cannot drop any of the above graphs for our purpose. Moreover, given a graph G we can, in randomized polynomial time, find either an embedding of one of these minors or conclude that treedepth of G is at most k c .This result has potential applications in a variety of settings where bounded treedepth plays a role. In addition to some graph structural applications, we describe a surprising application in circuit complexity and finite model theory from recent work of the second author [28].

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“…In [9], Dvořák, Giannopoulou, and Thilikos initiated the study of critical graphs having small tree-depth. They determined all k-critical graphs for k ≤ 4 and exhibited a construction of critical graphs from smaller ones; this construction is sufficient to construct all critical trees of any tree-depth.…”

Section: Introductionmentioning

confidence: 99%

On 1-uniqueness and dense critical graphs for tree-depth

Barrus

Sinkovic

2018

Discrete Mathematics

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The tree-depth of G is the smallest value of k for which a labeling of the vertices of G with elements from {1, . . . , k} exists such that any path joining two vertices with the same label contains a vertex having a higher label. The graph G is k-critical if it has tree-depth k and every proper minor of G has smaller tree-depth.Motivated by a conjecture on the maximum degree of k-critical graphs, we consider the property of 1-uniqueness, wherein any vertex of a critical graph can be the unique vertex receiving label 1 in an optimal labeling. Contrary to an earlier conjecture, we construct examples of critical graphs that are not 1-unique and show that 1-unique graphs can have arbitrarily many more edges than certain critical spanning subgraphs. We also show that (n − 1)-critical graphs are 1-unique and use 1-uniqueness to show that the Andrásfai graphs are critical with respect to tree-depth.

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Forbidden graphs for tree-depth

Cited by 33 publications

References 11 publications

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

A Polynomial Excluded-Minor Approximation of Treedepth

On 1-uniqueness and dense critical graphs for tree-depth

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