• Corpus ID: 232478913

Graph of uv-paths in 2-connected graphs

@inproceedings{Campo2021GraphOU,
  title={Graph of uv-paths in 2-connected graphs},
  author={Eduardo Rivera Campo},
  year={2021}
}
For a 2-connected graph G and vertices u, v of G we define an abstract graph P(Guv) whose vertices are the paths joining u and v in G, where paths S and T are adjacent if T is obtained from S by replacing a subpath Sxy of S with an internally disjoint subpath Txy of T . We prove that P(Guv) is always connected and give a necessary and a sufficient condition for connectedness in cases where the cycles formed by the replacing subpaths are restricted to a specific family of cycles of G. 
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By Theorem 6, C is ∆ * -dense and by Theorem 4, P C (G uv ) is connected
    Collorary 1. Let u and v be vertices of a 2-connected plane graph G. If C is the set of internal faces of G, then P C (G uv ) is connected
      Let u and v be vertices of a 2-connected graph G. If C is the set of cycles of G
        of G, then C is ∆ * -dense. We end this section with the following immediate corollaries