Characterisations and Linear-Time Recognition of Probe Cographs

Lê, Van Bang; Ridder, H. N. de

doi:10.1007/978-3-540-74839-7_22

Cited by 13 publications

(4 citation statements)

References 18 publications

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“…We review some results from [11,17]. (2) The class of probe cographs is hereditary and has finitely many bounds.…”

Section: Cographs and Related Notionsmentioning

confidence: 99%

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Induced betweenness in order-theoretic trees

Courcelle

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x<y<z∨z<y<x, and the corresponding betweenness structure is (N,B). The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, B(x,y,z) means that x<y<z or z<y<x or x<y≤x⊔z or z<y≤x⊔z, where x⊔z is the least upper-bound of incomparable elements x and z. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.

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“…We review some results from [11,17]. (2) The class of probe cographs is hereditary and has finitely many bounds.…”

Section: Cographs and Related Notionsmentioning

confidence: 99%

“…A bound of such a class C is a minimal induced subgraph that is not in C, a terminology used by Pouzet [20]. Partitioned probe cographs have six bounds, determined in [17].…”

Section: Introductionmentioning

confidence: 99%

Induced betweenness in order-theoretic trees

Courcelle

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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“…Because of the difficulty of the non-partitioned problem, many people have turned their attention to the recognition of probe interval graphs from specific families of graphs. Some examples of probe graph classes with non-partitioned recognition algorithms are chordal graphs [6], probe distancehereditary graphs [7], probe cographs [8], and probe comparability graphs [9]. In this paper, we will add to this list, giving an efficient non-partitioned recognition algorithm for probe interval 2-trees.…”

Section: Introductionmentioning

confidence: 99%

Recognition Algorithm for Probe Interval 2-Trees

Flesch

Nabity

2016

BJMCS

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Recognition of probe interval graphs has been studied extensively. Recognition algorithms of probe interval graphs can be broken down into two types of problems: partitioned and non-partitioned. A partitioned recognition algorithm includes the probe and nonprobe partition of the vertices as part of the input, where a non-partitioned algorithm does not include the partition. Partitioned probe interval graphs can be recognized in linear-time in the edges, whereas non-partitioned probe interval graphs can be recognized in polynomial-time. Here we present a non-partitioned recognition algorithm for 2-trees, an extension of trees, that are probe interval graphs. We show that this algorithm runs in O(m) time, where m is the number of edges of a 2-tree. Currently there is no algorithm that performs as well for this problem.

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“…Several partitioned probe problems have been studied, all of them so far classified as polynomial: cographs [12], P 4 -parse [10], permutation graphs [11], threshold [1], chordal graphs [2], chain graphs [7], and trivially perfect graphs [1], leading to the Probe Graph Conjecture (PGC): "Partitioned probe graphs of C are polynomially recognizable whenever C is polynomially recognizable" [12]. More results and open problems on partitioned probe graphs can be found in [3].…”

Section: Introductionmentioning

confidence: 99%

Searching for a NP-Complete Probe Graph Problem

TEIXEIRA,

de Figueiredo,

Dantas

2012

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A probe graph for a graph class C is a graph G = (V, E), such that it is possible to add some edges between vertices of an independent set N in order to obtain a graph G ′ belonging to C [7]. If the independent set N ⊂ V is given as input, we have a special case of graph sandwich problem [6], called partitioned probe problem. A graph partition of a graph G = (V, E) is a partition of V into a number of parts. A graph partition problem consists in finding a graph partition where the parts satisfy some internal or external constraints. A three nonempty part problem is a graph partition problem, such that V must be partitioned in exactly three nonempty parts. All possible such nonempty part problems, reflecting the various combinations of internal and external constraints, are classified as polynomial or NPcomplete in both recognition and sandwich versions [14]. We focus on three nonempty part partitioned probe problems, for which their sandwich version is NP-complete and their recognition version is polynomial. We show that most of those partitioned probe problems have a behavior strongly similar to their recognition version, despite

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Characterisations and Linear-Time Recognition of Probe Cographs

Cited by 13 publications

References 18 publications

Induced betweenness in order-theoretic trees

Induced betweenness in order-theoretic trees

Recognition Algorithm for Probe Interval 2-Trees

Searching for a NP-Complete Probe Graph Problem

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