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2.5-Connectivity: Unique Components, Critical Graphs, and Applications

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Graph-Theoretic Concepts in Computer Science (WG 2020)

Abstract

If a biconnected graph stays connected after the removal of an arbitrary vertex and an arbitrary edge, then it is called 2.5-connected. We prove that every biconnected graph has a canonical decomposition into 2.5-connected components. These components are arranged in a tree-structure. We also discuss the connection between 2.5-connected components and triconnected components and use this to present a linear time algorithm which computes the 2.5-connected components of a graph. We show that every critical 2.5-connected graph other than \(K_4\) can be obtained from critical 2.5-connected graphs of smaller order using simple graph operations. Furthermore, we demonstrate applications of 2.5-connected components in the context of cycle decompositions and cycle packings.

This research was partially funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148), and the Federal Ministry of Education and Research.

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Notes

  1. 1.

    This differs from the definition of 2-connected graphs as can be found in  [3]. Connected graphs of order 2 are biconnected but not 2-connected.

  2. 2.

    Recall the definition of \(S(\mathcal {I})\) from Lemma 1.

  3. 3.

    Recall the definition of \(S(\mathcal {I})\) from Lemma 1.

  4. 4.

    Originally, Hajós conjectured that at most \(\nicefrac {1}{2}|V(G)|\) cycles are needed. This equivalent reformulation is due to Fan and Xu, cf.  [4].

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Heinrich, I., Heller, T., Schmidt, E., Streicher, M. (2020). 2.5-Connectivity: Unique Components, Critical Graphs, and Applications. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_28

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  • DOI: https://doi.org/10.1007/978-3-030-60440-0_28

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