Gonality of complete graphs with a small number of omitted edges

Let K d be the complete metric graph on d vertices. We compute the gonality of graphs obtained from K d by omitting edges forming a K h , or general configurations of at most d − 2 edges. We also investigate if these graphs can be lifted to curves with the same gonality. We lift the former graphs and the ones obtained by removing up to d − 2 edges not forming a K 3 using models of plane curves with certain singularities. We also study the gonality when removing d − 1 edges not forming a K 3 . We use harmonic morphism to lift these graphs to curves with the same gonality because in this case plane singular models can no longer be used due to a result of Coppens and Kato.