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arXiv · 2003.01498

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

Irene Heinrich·Till Heller·Eva Schmidt·Manuel Streicher
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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 K4 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.

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Irene Heinrich, Till Heller, Eva Schmidt, Manuel Streicher. 2020-03-03. 2.5-Connectivity: Unique Components, Critical Graphs, and Applications. https://arxiv.org/abs/2003.01498

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