On the extension of bipartite to parity graphs

Cicerone, Serafino; Stefano, Gabriele Di

doi:10.1016/s0166-218x(99)00074-8

Cited by 24 publications

(22 citation statements)

References 17 publications

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“…Cograph [23] Distance-Hereditary [1] Switch Cograph [14] (6, 2)-Chordal Bipartite [9] Parity [10] Alternatively, a distance-hereditary graph can be viewed as a graph that is constructed using some sequence 2 of pendant, strong twin, and weak twin operations, which was first introduced by Bandelt and Mulder [1]. DEFINITION 2.1.…”

Section: Characterization Of Certain Classes Of Graphsmentioning

confidence: 99%

“…To do so, we use a tree representation of cographs, known as cotrees, which was first introduced by Corneil et al [11], and which uniquely represents cographs 10 . DEFINITION 4.2.…”

Section: Exponential Bounds Through Bicolored Cographsmentioning

confidence: 99%

“…10 These trees are exactly the same as the vertex incremental trees for cographs later developed by Nakano et al [23]. 11 Note that when we use SET in our grammar, we are simply indicating that there is no order to the children.…”

Section: Exponential Bounds Through Bicolored Cographsmentioning

confidence: 99%

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Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

Lumbroso¹,

Shi²

2018

2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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In this paper, building on previous work by Nakano et al.[23], we develop an alternate technique which almost automatically translates (existing) vertex incremental characterizations of graph classes into asymptotics of that class. Specifically, we construct trees corresponding to the sequences of vertex incremental operations which characterize a graph class, and then use analytic combinatorics to enumerate the trees, giving an upper bound on the graph class. This technique is applicable to a wider set of graph classes compared to the tree decompositions, and we show that this technique produces accurate upper bounds.We first validate our method by applying it to the case of distance-hereditary graphs, and comparing the bound obtained by our method with that obtained by Nakano et al. [23], and the exact enumeration obtained by Chauve et al. [7,8]. We then illustrate its use by applying it to switch cographs, for which there are few known results: our method provide a bound of ∼ 6.301 n

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Section: Characterization Of Certain Classes Of Graphsmentioning

confidence: 99%

“…To do so, we use a tree representation of cographs, known as cotrees, which was first introduced by Corneil et al [11], and which uniquely represents cographs 10 . DEFINITION 4.2.…”

Section: Exponential Bounds Through Bicolored Cographsmentioning

confidence: 99%

See 1 more Smart Citation

Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

Lumbroso¹,

Shi²

2018

2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…1-Join decompositions (also known as split decompositions) were used for circle graph recognition [71,116] and parity graph recognition [40,60]. Cunningham [58] showed how 1-join decompositions can be used for the independent set problem, and Cicerone and Di Stefano [40] showed how this algorithm can be applied to parity graphs.…”

Section: -Joinsmentioning

confidence: 99%

“…Cunningham [58] showed how 1-join decompositions can be used for the independent set problem, and Cicerone and Di Stefano [40] showed how this algorithm can be applied to parity graphs. Rao [106] shows how to use 1-join decompositions for the clique and coloring problems.…”

Section: -Joinsmentioning

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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Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition

Dahlhaus¹

2000

Journal of Algorithms

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On the extension of bipartite to parity graphs

Cited by 24 publications

References 17 publications

Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

The world of hereditary graph classes viewed through Truemper configurations

Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition

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