Relative numbers of ends and quasi-median graphs

Given a finitely generated $G$ and a subgraph $H\le G$, the relative number of ends $e(G,H)$ is the number of ends of a Schreier graph $\mathrm{S}\mathrm{c}\mathrm{h}(G,H)$ and the number of coends $\tilde{e}(G,H)$ is the maximal number of $H$-infinite components of the complement of a neighbourhood of $H$ in $G$. Generalising Sageev’s characterisation of codimension-one subgroups in terms of actions on CAT(0) cube complexes, we characterise the number of relative ends and the number of coends of a pair $(G,H)$ in terms of actions on quasi-median graphs.