I am currently interested in learning more about perfect graphs, and perfectly orderable graphs, however some of the basic concepts are escaping me, and I am simply unable to wrap my head around some of the definitions and examples I am facing. I understand that, for example, a perfectly orderable graph is one where its vertices may be ordered in such a way that allows it to be optimally colored via a greedy coloring algorithm - https://en.wikipedia.org/wiki/Perfectly_orderable_graph.
Here is another definition:
"Given an ordered graph (G, < ), the ordering < is called perfect (Chvátal, V., Perfectly ordered graphs, Perfect graphs, Ann. Discrete Math. 21, 63-65 (1984). ZBL0559.05055.) if for each induced ordered subgraph (H, < ) the greedy algorithm produces an optimal colouring of H. The graphs admitting a perfect ordering are called perfectfy orderable. An obstruction in an ordered graph is a chordless path with four vertices abcd such that a <b and d <c. It is easily seen that a perfectly ordered graph has no obstruction. Chvatal has shown that this condition is also sufficient: a graph is perfectly orderable if and only if it admits an obstruction-free ordering" (Hoàng, Chín T.; Maffray, Frédéric; Olariu, Stephan; Preissmann, Myriam, A charming class of perfectly orderable graphs, Discrete Math. 102, No. 1, 67-74 (1992). ZBL0776.05091.)
Nevertheless, I am struggling to understand how to determine that a graph is perfectly orderable (or not perfectly orderable). For example a perfectly orderable graph would be (with a perfect ordering, for example of c<d<e<b<a, and the only two induced subgraphs of 4 vertices, being: abcd & aedc):
In short, why is the first one perfectly orderable but not the other?