# The difference between perfect Perfectly orderable graph and Perfect Elimination Ordering?

I am currently trying to learn in depth different concepts in graph theory. I can not quite understand what the difference is between Perfectly orderable graph and Perfect Elimination Ordering?

I would be happy to help with this. Thanks

• May 10, 2021 at 10:15

A vertex order $$<$$ of $$G$$ is perfect if and only if $$G$$ contains no $$P_4$$ $$abcd$$ with $$a < b$$ and $$d < c$$.
A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex $$v,$$ $$v$$ and the neighbors of $$v$$ that occur after $$v$$ in the order form a clique. Explained/simpler definition.