Detecting a Theta or a Prism

We show that there is a polynomial time algorithm that, given three vertices of a graph, tests whether there is an induced subgraph that is a tree, containing the three vertices. (Indeed, there is an explicit construction of the cases when there is no such tree.) As a consequence, we show that there is a polynomial time algorithm to test whether a graph contains a "theta" as an induced subgraph (this was an open question of interest) and an alternative way to test whether a graph contains a "pyramid" (a fundamental step in checking whether a graph is perfect).

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“…Motivated by the parallel between pyramids and thetas-or-prisms, Chudnovsky and Kapadia [2,4] proved the following:…”

Section: 1mentioning

confidence: 99%

The three-in-a-tree problem

Chudnovsky

Seymour

2010

Combinatorica

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“…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]. The high complexity of all these algorithms is due to the cleaning procedure.…”

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

confidence: 99%

“…Note that the condition that nodes of P i ∪ P j , i = j, must induce a hole, implies that all paths of a 3P C(·, ·) have length greater than one, and at most one path of a 3P C(∆, ·) has length one. In literature 3P C(·, ·) is also referred to as theta [23], 3P C(∆, ·) as pyramid [22], and 3P C(∆, ∆) as prism [23].…”

Section: Introductionmentioning

confidence: 99%

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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“…Note that the condition that nodes of P i ∪ P j , i = j, must induce a hole, implies that all paths of a 3P C(·, ·) have length greater than one, and at most one path of a 3P C(∆, ·) has length one. 3P C(·, ·)'s are also known as thetas (as in [8]), 3P C(∆, ∆)'s are also known as prisms (as in [8]), and 3P C(∆, ·)'s are also known as pyramids (as in [7]). …”

Section: Figure 1: Truemper Configurationsmentioning

confidence: 99%

“…This turns out to be false so far. Detecting a theta or a prism using the method outlined in [9] ends up being of complexity O(n 35 ) [8].…”

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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Detecting a Theta or a Prism

Cited by 25 publications

References 3 publications

The three-in-a-tree problem

The three-in-a-tree problem

The world of hereditary graph classes viewed through Truemper configurations

Even-hole-free graphs: A survey

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