Relating Different Cycle Spaces of the Same Infinite Graph

Abstract

Casteels and Richter have shown that if $X$ and $Y$ are distinct compactifications of a locally finite graph $G$ and $f:X\to Y$ is a continuous surjection such that $f$ restricts to a homeomorphism on $G$, then the cycle space $Z_X$ of $X$ is contained in the cycle space $Z_Y$ of $Y$. In this work, we show how to extend a basis for $Z_X$ to a basis of $Z_Y$.