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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

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Standard proof that cyclic ordering of edges is preserved under planar graph homotopy?

I have several questions about the following theorem statement: Thm: Let $G = (V, E)$ be a planar graph, and let $\varphi_0 : G \rightarrow \mathbb{R}^2$, $\varphi_1 : G \rightarrow \mathbb{R}^2$ be …