Question $1$: Suppose that a connected planar graph has $50$ edges. If a planar representation of this graph divides the plane into $30$ regions, how many vertices does this graph have?

Question $2$: Determine whether the given graph has an Euler circuit. Construct such a circuit when one exists. If no Euler circuit exists, determine whether the graph has an Euler path and construct such a path if one exists.

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