Locally finite graphs with an excluded finite minor have the weak coarse Menger property with f depending only on k and g linear in r independent of k.
Sachs,On spatial representation of finite graphs, Proceedings of a conference held in Lag´ ow, Poland, February 10–13, 1981, Lecture Notes in Mathematics, Vol
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Coarse Menger property of quasi-minor excluded graphs and length spaces
Locally finite graphs with an excluded finite minor have the weak coarse Menger property with f depending only on k and g linear in r independent of k.