You have discovered an old paper on graph theory that discusses the viscosity of a graph (which for all you know, is something completely made up by the author). A theorem in the paper claims that “if a graph satisfies condition (V), then the graph is viscous.” Which of the following are equivalent ways of stating this claim? Which are equivalent to the converse of the claim?
  1. A graph is viscous only if it satisfies condition (V).
  2. A graph is viscous if it satisfies condition (V).
  3. For a graph to be viscous, it is necessary that it satisfies condition (V).
  4. For a graph to be viscous, it is sufficient for it to satisfy condition (V).
  5. Satisfying condition (V) is a sufficient condition for a graph to be viscous.
  6. Satisfying condition (V) is a necessary condition for a graph to be viscous.
  7. Every viscous graph satisfies condition (V).
  8. Only viscous graphs satisfy condition (V).