Graph of $uv$-Paths in Connected Graphs

Discrete and Computational Geometry, Graphs, and Games

Abstract

For a 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 $T$. Let $\mathcal{C}$ be a set of cycles of $G$; the \emph{$uv$-path graph of $G$ defined by $\mathcal{C}$} is the spanning subgraph $\mathcal{P}_\mathcal{C}(G_{uv})$ of $\mathcal{P}(G_{uv})$ in which two paths $S$ and $T$ are adjacent if and only if the unique cycle $\sigma$ contained in $S\cup T$ lies in $\mathcal{C}$. We prove that $\mathcal{P}(G_{uv})$ is always connected and give a necessary condition and a sufficient condition for a graph $\mathcal{P}_\mathcal{C}(G_{uv})$ to be connected.