Graph of uv-paths in 2-connected graphs

TL;DR

This paper introduces a new graph structure based on $u$-$v$ paths in 2-connected graphs, proving its universal connectivity and exploring conditions for connectedness under cycle restrictions.

Contribution

It defines the graph $ ext{P}(G_{uv})$ of $u$-$v$ paths, proves its connectivity universally, and characterizes connectedness under certain cycle constraints.

Findings

$ ext{P}(G_{uv})$ is always connected for 2-connected graphs.

Provides necessary and sufficient conditions for connectedness with cycle restrictions.

Establishes a new framework for analyzing path transformations in graphs.

Abstract

For a $2$-connected graph $G$ and vertices $u,v$ of $G$ we define an abstract graph $\mathcal{P}(G_{uv})$ 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 $S_{xy}$ of $S$ with an internally disjoint subpath $T_{xy}$ of $T$. We prove that $\mathcal{P}(G_{uv})$ 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 $G$.