We consider a variant of metrised graphs where the edge lengths take values in a commutative monoid, as a higher-rank generalisation of the notion of a tropical curve. Divisorial gonality, which Baker and Norine defined on combinatorial graphs in terms of a chip firing game, is extended to these monoid-metrised graphs. We define geometric gonality of a monoid-metrised graph as the minimal degree of a horizontally conformal, non-degenerate morphism onto an monoid-metrised tree, and prove that geometric gonality is an upper bound for divisorial gonality in the monoid-metrised case. We also show the existence of a subdivision of the underlying graph whose gonality is no larger than the monoid-metrised gonality. Finally, we relate our notion of geometric gonality to the minimal degree of a map between logarithmic curves.

JvDdB was partially supported by the Dutch Research Council (NWO), project number 613.009.127. DH was partially supported by the Dutch Research Council (NWO), project number VI.Vidi.193.006.