Abstract
An edge e of a graph G will be said to be genus-minimal if \(\gamma \left( G-e \right)<\gamma \left( G \right)or\widetilde{\gamma }\left( G-e \right)<\widetilde{\gamma }(G)\) where γ and \(\widetilde{\gamma }\) denote the genus and the crosscap number ( = nonorientable genus), respectively. A graph G will be called minimal with respect to a closed surface S if G does not embed on S but every proper subgraph of G embeds on S. In particular, every edge of a minimal graph ( with respect to some closed surface) is genus-minimal.
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