I'm reading the Extremal Graph Theory book by Bollobás, and I'm stuck at the definition of 'compatible vertices'.
It's here at the bottom of p.13
It says : "Call two vertices compatible if every edge joining two vertices of any path connecting them belongs to the path."
My reaction to this is that if an edge joining two vertices is on a given path, how could it NOT be on the path ? Can someone provide an intuition of compatible vertices ? And give an example of vertices that are not compatible ?
"Call two vertices compatible if every edge joining two vertices of any path connecting them belongs to the path." So if two vertices are adjacent, but there exists a path through them that doesn't use the edge joining them, then the vertices on either end of said path are not compatible.
The edge you're thinking about isn't on the given path. The vertices are. The edge is on a different path.
Please accept this humble MSPaint drawing.
You can see the vertices on opposite ends wouldn't be considered compatible, because edge 4 joins two vertices on the green path, but it is not on the green path.
Just for completeness' sake, here's an example of when the vertices are compatible.
You can see how this fits with the concept of "minimally connected", which the book is discussing in that section.