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.
References
SHOWING 1-9 OF 9 REFERENCES
On the perfect matching graph defined by a set of cycles
- Mathematics
- 2017
The perfect matching graph of a graph G, denoted by M(G), has one vertex for each perfect matching of G and two matchings are adjacent if their symmetric difference is a cycle of G. Let C be a family…
Monotone paths on polytopes
- Mathematics
- 2000
Abstract. We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytopeP with respect to a generic functionalf. The third author has conjectured that this graph is always (d…
Hamilton Circuits in Tree Graphs
- Mathematics, Computer Science
- 1966
It is shown that any tree graph containing more than two vertices has Property H, and two operations for augmenting networks (linear graphs) are defined: edge insertion and vertex insertion.
Enumerative problems for arborescences and monotone paths on polytopes
- Mathematics
- 2020
Every generic linear functional $f$ on a convex polytope $P$ orients the edges of the graph of $P$. In this directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$.…
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