Graph Extremities Defined by Search Algorithms

Berry, Alain; Blair, Jean R. S.; Bordat, Jean Paul; Simonet, Geneviève

doi:10.3390/a3020100

Cited by 22 publications

(33 citation statements)

References 21 publications

Supporting

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“…We show in this paper that a slight variant of MLS computes a PMO of the input chordal graph for each labeling structure. As this variant is equivalent to MLS if the order on the labels is total, this generalizes the result from [12] that MLS used with totally ordered labels computes a PMO of a chordal graph.…”

Section: Introductionsupporting

confidence: 66%

“…Note that the definition of a labeling structure given in [12] is less general than the definition given in this paper, but the proof of this result still holds here. We define the algorithm moplex-MLS (Algorithm 5), which computes a PMO of a chordal graph, whether the order on labels is total or not, by adding, in the case where the ordering fails to be total, a tie-breaking rule for choosing a vertex with a maximal label.…”

Section: Clique Tree Using Mlsmentioning

confidence: 78%

“…The algorithms MCS, LexBFS, LexDFS and, more generally, the algorithm L-MLS for any labeling structure L for which the order on labels is total compute a PMO of a connected chordal graph [12].…”

Section: Clique Tree Using Mlsmentioning

confidence: 99%

“…We thus provide an alternate proof for the result from [12] that if the order on labels is total, then each L-MLS ordering of a chordal graph is a PMO of this graph, with a more general definition of a labeling structure and an alternative (simpler) proof.…”

Section: Lemmamentioning

confidence: 99%

“…Berry and Bordat [8] showed that LexBFS ends on a moplex of the input graph whether it is chordal or not, which implies that LexBFS computes a PMO of the input graph if it is chordal. Berry, Blair, Bordat and Simonet [12] proved the more general result that any instance of the algorithm MLS with totally ordered labels computes a PMO of a chordal graph. Berry and Pogorelcnik [13] showed that MCS and LexBFS can be modified to compute the minimal separators and the maximal cliques of a chordal graph, using the fact that the computed ordering is a PMO of this graph, thus extending the result for MCS from [4].…”

Section: Introductionmentioning

confidence: 99%

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Computing a Clique Tree with the Algorithm Maximal Label Search

Berry

Simonet

2017

Algorithms

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Abstract:The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.

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Section: Introductionsupporting

confidence: 66%

Section: Clique Tree Using Mlsmentioning

confidence: 78%

Section: Clique Tree Using Mlsmentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing a Clique Tree with the Algorithm Maximal Label Search

Berry

Simonet

2017

Algorithms

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Avoidable Vertices and Edges in Graphs

Beisegel

Chudnovsky

Gurvich

et al. 2019

Lecture Notes in Computer Science

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A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF -vertices), simplicial paths, and their common generalization avoidable paths, introduced here. We present a general conjecture on the existence of avoidable paths. If true, the conjecture would imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices, and a result due to Chvátal, Sritharan, and Rusu from 2002 the existence of simplicial paths. In turn, both of these results generalize Dirac's classical result on the existence of simplicial vertices in chordal graphs.We prove that every graph with an edge has an avoidable edge, which settles the first open case of the conjecture. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the proved cases of the conjecture have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle. * An extended abstract of this work is

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End-Vertices of AT-free Bigraphs

Gorzny

Huang

2020

Lecture Notes in Computer Science

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Graph Extremities Defined by Search Algorithms

Cited by 22 publications

References 21 publications

Computing a Clique Tree with the Algorithm Maximal Label Search

Computing a Clique Tree with the Algorithm Maximal Label Search

Avoidable Vertices and Edges in Graphs

End-Vertices of AT-free Bigraphs

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