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Is there a disconnected graph with degree sequence $(4$, $4$, $3, 3, 3, 3, 3, 3)$?

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    By "disconnected", do you mean to indicate that the graph certainly has at least two components, or do you instead mean "possibly disconnected"?2012-11-09

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One component must have at least 5 vertices. Any other must have at least 4. But you only have 8 vertices.