The Recognition Problem of Graph Search Trees

Beisegel, Jesse; Denkert, Carolin; Köhler, Ekkehard; Krnc, Matjaž; Pivač, Nevena; Schleicher, Robert; Strehler, Martin

doi:10.1137/20m1313301

Cited by 8 publications

(9 citation statements)

References 23 publications

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“…In the graph depicted in Figure 2(b), every vertex is avoidable, but neither x 1 nor x 2 can be the end vertex of an MCS. Furthermore, just as in the case of LBFS, deciding whether a vertex is an end vertex of MCS is NP-complete [5].…”

Section: Computing Avoidable Verticesmentioning

confidence: 99%

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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Section: Computing Avoidable Verticesmentioning

confidence: 99%

Avoidable Vertices and Edges in Graphs

Beisegel

Chudnovsky

Gurvich

et al. 2019

Lecture Notes in Computer Science

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“…We first show that LexBFS and LexDFS have the same set of L-trees on chordal graphs. This fact is also implied by a more general result in [1]. Since this work has not been published yet and we only need a special case here, we give an alternative proof for the sake of completeness.…”

Section: Lexdfs On Chordal Graphsmentioning

confidence: 58%

“…Note that this result does not hold for search orders and their corresponding trees in general (if P = N P). Beisegel et al [1,2] show for example that the recognition problem of F-trees of both LexBFS and LexDFS is N P-complete, whereas it is easy to recognize the corresponding orders.…”

Section: There Is An Algorithm With Running Time In O(m ) For Any Gra...mentioning

confidence: 99%

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Linear Time LexDFS on Chordal Graphs

Beisegel,

Köhler,

Scheffler

et al. 2020

Preprint

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Lexicographic Depth First Search (LexDFS) is a special variant of a Depth First Search (DFS), which was introduced by Corneil and Krueger in 2008. While this search has been used in various applications, in contrast to other graph searches, no general linear time implementation is known to date. In 2014, Köhler and Mouatadid achieved linear running time to compute some special LexDFS orders for cocomparability graphs. In this paper, we present a linear time implementation of LexDFS for chordal graphs. Our algorithm is able to find any LexDFS order for this graph class. To the best of our knowledge this is the first unrestricted linear time implementation of LexDFS on a non-trivial graph class. In the algorithm we use a search tree computed by Lexicographic Breadth First Search (LexBFS).

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“…Furthermore, they extended these results to several graph classes. Recently, Beisegel et al [2] proved N P-hardness results for MCS and MNS, and they also provided linear time algorithms for this problem on split graphs and unit interval graphs.…”

Section: F-tree (L-tree) Recognition Problemmentioning

confidence: 99%

Recognizing Graph Search Trees

Beisegel

Denkert

Köhler

et al. 2019

Electronic Notes in Theoretical Computer Science

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Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show N P-completeness. We complement our results by providing linear time algorithms for these searches on split graphs. * The work of this paper was done in the framework of a bilateral project between Brandenburg

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The Recognition Problem of Graph Search Trees

Cited by 8 publications

References 23 publications

Avoidable Vertices and Edges in Graphs

Avoidable Vertices and Edges in Graphs

Linear Time LexDFS on Chordal Graphs

Recognizing Graph Search Trees

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