Meyniel graph

[1] The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them.

The Meyniel graphs are named after Henri Meyniel (also known for Meyniel's conjecture), who proved that they are perfect graphs in 1976,[2] long before the proof of the strong perfect graph theorem completely characterized the perfect graphs.

The same result was independently discovered by Markosjan & Karapetjan (1976).

Thus, the Meyniel graphs meet the definition of being a perfect graph, that the clique number equals the chromatic number in every induced subgraph.

[1][2][3] Meyniel graphs are also called the very strongly perfect graphs, because (as Meyniel conjectured and Hoàng proved) they can be characterized by a property generalizing the defining property of the strongly perfect graphs: in every induced subgraph of a Meyniel graph, every vertex belongs to an independent set that intersects every maximal clique.

In a Meyniel graph, every long odd cycle (such as the black 5-cycle shown here) must have at least two chords (green)
The house graph is perfect but not Meyniel