The three-in-a-tree problem

Chudnovsky, Maria; Seymour, Paul

doi:10.1007/s00493-010-2334-4

Cited by 70 publications

(97 citation statements)

References 4 publications

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“…These problems are closely related to each other and to the k-Induced Disjoint Paths problem. For general graphs, even the problems 2-in-a-Cycle and 3-in-a-Path are NP-complete [2,12], whereas the k-in-a-Tree problem is polynomial-time solvable for k = 3 [6], open for any fixed k ≥ 4, and NP-complete when k is part of the input [8]. Several polynomial-time solvable cases are known for graph classes, see e.g.…”

Section: Related Problemsmentioning

confidence: 99%

Induced Disjoint Paths in Claw-Free Graphs

Golovach¹,

Paulusma²,

Leeuwen³

2015

SIAM J. Discrete Math.

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Abstract. Paths P1, . . . , P k in a graph G = (V, E) are said to be mutually induced if for any 1 ≤ i < j ≤ k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si, ti) contains k mutually induced paths Pi such that Pi connects si and ti for i = 1, . . . , k. We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are proven to be fixed-parameter tractable for claw-free graphs as well. We show that an improvement of these results in certain directions is unlikely, for example by noting that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of clawfree graphs), unless NP ⊆ coNP/poly. Moreover, the problem becomes NP-complete, even when k = 2, for the more general class of K1,4-free graphs. Finally, we show that the n O(k) -time algorithm of Fiala et al. for testing whether a claw-free graph contains some k-vertex graph H as a topological induced minor is essentially optimal by proving that this problem is W[1]-hard even if G and H are line graphs.

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Section: Related Problemsmentioning

confidence: 99%

Induced Disjoint Paths in Claw-Free Graphs

Golovach¹,

Paulusma²,

Leeuwen³

2015

SIAM J. Discrete Math.

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“…Chudnovsky and Seymour [29] show that detecting whether a graph contains a 3P C(·, ·) can be done in O(n 11 ) time. For this detection problem, the shortest-paths detector technique does not work.…”

Section: Recognizing Truemper Configurationsmentioning

confidence: 99%

“…For this detection problem, the shortest-paths detector technique does not work. The detection of 3P C(·, ·)'s relies on being able to solve a more general problem called the three-in-a-tree problem defined as follows: given a graph G and three specified vertices a, b and c, the question is whether G contains a tree that passes through a, b and c. It is shown in [29] that this problem can be solved in O(n 4 ) time. What is interesting is that the algorithm for the three-in-a-tree problem is based on an explicit construction of the cases when the desired tree does not exist, and that this construction can be directly converted into the algorithm.…”

Section: Recognizing Truemper Configurationsmentioning

confidence: 99%

“…Finally, by using Theorem 9.3 Chang and Lu [11] obtain an O(n 11 ) recognition algorithm for even-hole-free graphs. They improve the complexity by introducing a new idea of a "tracker" that allows for fewer graphs that need to be recursively decomposed by star cutsets, and they improve the complexity of the cleaning procedure by first looking for certain structures, using the three-in-a-tree algorithms from [29], before applying the cleaning. We observe that detecting whether a graph contains a 3P C(·, ·) or a 3P C(∆, ∆) can be done in O(n 35 ) time [23].…”

Section: Theorem 91 ([43]) a Graph Is Odd-signable If And Only If Itmentioning

confidence: 99%

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The world of hereditary graph classes viewed through Truemper configurations

Vušković¹

2013

Surveys in Combinatorics 2013

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In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few specific types, that we will call Truemper configurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices).The configurations that Truemper identified in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper configurations, focusing on the algorithmic consequences, trying to understand what structurally enables efficient recognition and optimization algorithms.

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“…This algorithm first needs to test for 3P C(·, ·)'s (thetas) and 3P C(∆, ∆)'s (prisms) in that time. It turns out that testing for thetas can be done in time O(n 11 ) [11]. Detecting a prism is NP-complete in general [31].…”

Section: Recognition Algorithmsmentioning

confidence: 99%

Even-hole-free graphs: A survey

Vušković

2010

Appl Anal Discrete Math

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The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs, which was the key initial motivation for their study. The techniques developed in the study of even-hole-free graphs were generalized to other complex hereditary graph classes, and in the case of perfect graphs led to the famous resolution of the Strong Perfect Graph Conjecture and their polynomial time recognition. The class of even-holefree graphs is also of independent interest due to its relationship to β-perfect graphs. In this survey we describe all the different structural characterizations of even-hole-free graphs, focusing on their algorithmic consequences.

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The three-in-a-tree problem

Cited by 70 publications

References 4 publications

Induced Disjoint Paths in Claw-Free Graphs

Induced Disjoint Paths in Claw-Free Graphs

The world of hereditary graph classes viewed through Truemper configurations

Even-hole-free graphs: A survey

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