I'm self studying Graph Theory and its Applications by Jonathan Gross and Jay Yellen and ran into a proof I can't make sense of. I have the first edition, but I have checked later editions.
Here's the proof, and the particular problem I was having is in the second paragraph of the sufficiency section. Why we can conclude that $H \oplus C \oplus S$ contains $b$ and only $b$. In fact since $H \oplus C$ and $S$ contain $b$, shouldn't $b$ not be in $H \oplus C \oplus S$?
