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We prove the following "linkage" theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity.
Culler, M.; Vogtmann, K.: Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), no. 1, 91-119
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Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and Theta functions
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Mikhalkin, G.: Tropical Geometry
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Mikhalkin, G.: What is … \dots a tropical curve?
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Gathmann, A.; Kerber, M.; Markwig, H.: Tropical fans and the moduli spaces of tropical curves
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