Take a genus $g$ surface $S$ standardly embedded in $\mathbb{R}^3$, by which I mean it is unknotted. Surface $S$ bounds a volume $V$ that deformation retracts on a standardly embedded planar graph $G$ with $\beta\_1 = g$, and that only has degree $3$ vertices. 


Among the loops on $S$ that are null homotopic in $V$, there is a subset that are boundaries of embedded disks in $V$ that intersect $G$ exactly once for some choice of $G$ as above.


Do these loops (or perhaps close variants) have a name? Do they have an alternate definition?



"Basic" loops on standardly embedded surfaces
They are often called meridians of $G$. Note that there are many graphs $G$ to which $V$ deformation retracts (most nonplanar); if you are not particular about which graph $G$ then they are called meridians of $V$. $V$ is called a handlebody and $G$ is a spine of the handlebody. See, for example, Scharlemann's "Refilling meridians in a genus 2 handlebody complement" [arXiv:math/0603705](https://arxiv.org/abs/math/0603705)


