GKM actions on cohomogeneity one manifolds

We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group diagram of $M$ such that the $T$-action on $M$ is of GKM type, and describe its GKM graph. The general results are illustrated on explicit examples.
18 pages, 6 figures, 3 tables. Minor changes, in particular additional explanations were added in the proof of Proposition 2.3. Proposition 3.6 added. Results are unchanged