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Is there any reason why, for example, in the theorems about embeddings of graphs, at some point the authors states that all graphs are connected.

To be more concrete, after prop 3.1, and 3.2 in this paper, the author says "we shall always assume that graphs are connected."

Why? is this just for simplicity? Or what if we consider disconnected graphs? Can these graphs also be embedded in surfaces?

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It is probably just for simplicity. Many results in graph theory apply to each connected component individually, so connectivity is a common simplifying assumption (without loss of generality). For example, a graph is $k$-colorable if and only if each component is $k$-colorable.

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