bounded components

• assumed
  • graph class edgeless has constant bounded components – By definition
  • bounded components upper bounds maximum degree by a linear function – By definition
• unknown source
  • bounded components upper bounds distance to bounded components by a linear function – by definition
  • bounded components upper bounds cutwidth by a polynomial function – By greedily placing one component after another.
  • bounded bounded components does not imply bounded distance to perfect – By a disjoint union of small components with distance to perfect at least 1.
  • bounded bounded components does not imply bounded distance to planar – By a disjoint union of many $K_5$ graphs.