Gallai’s Path Decomposition for Planar Graphs

Blanché, Alexandre; Bonamy, Marthe

doi:10.1007/978-3-030-83823-2_121

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…This would answer a question posed by Erdős. Both are posed in [27]. The conjecture remains open in general, but it has been proven for many classes of graphs, such as graphs whose even-degree vertices induce a forest [30], planar graphs [6], graphs with maximum degree at most 5 [7], and graphs with maximum degree 6 in which the vertices of maximum degree form an independent set (with some exceptions) [13]. A number of related invariants have been introduced in the study of Gallai's conjecture.…”

Section: Introductionmentioning

confidence: 99%

Shortest-Path Kernels on Graphs

Borgwardt

Kriegel

Fifth IEEE International Conference on Data Mining (ICDM'05)

635

543

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Data mining algorithms are facing the challenge to deal with an increasing number of complex objects. For graph data, a whole toolbox of data mining algorithms becomes available by defining a kernel function on instances of graphs. Graph kernels based on walks, subtrees and cycles in graphs have been proposed so far. As a general problem, these kernels are either computationally expensive or limited in their expressiveness. We try to overcome this problem by defining expressive graph kernels which are based on paths. As the computation of all paths and longest paths in a graph is NP-hard, we propose graph kernels based on shortest paths. These kernels are computable in polynomial time, retain expressivity and are still positive definite. In experiments on classification of graph models of proteins, our shortest-path kernels show significantly higher classification accuracy than walk-based kernels.

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Section: Introductionmentioning

confidence: 99%

Shortest-Path Kernels on Graphs

Borgwardt

Kriegel

Fifth IEEE International Conference on Data Mining (ICDM'05)

635

543

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Separating path systems of almost linear size

Letzter

2024

Trans. Amer. Math. Soc.

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A separating path system for a graph G G is a collection P \mathcal {P} of paths in G G such that for every two edges e e and f f , there is a path in P \mathcal {P} that contains e e but not f f . We show that every n n -vertex graph has a separating path system of size O ( n log ∗ ⁡ n ) O(n \log ^* n) . This improves upon the previous best upper bound of O ( n log ⁡ n ) O(n \log n) , and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an O ( n ) O(n) bound should hold.

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Towards the Erdős-Gallai Cycle Decomposition Conjecture

Bucić

Montgomery

2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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Gallai’s Path Decomposition for Planar Graphs

Cited by 3 publications

References 7 publications

Shortest-Path Kernels on Graphs

Shortest-Path Kernels on Graphs

Separating path systems of almost linear size

Towards the Erdős-Gallai Cycle Decomposition Conjecture

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