A Meyniel graph, also called a very strongly perfect graph, is a graph in which every odd graph cycle of length five or more has at least two chords. Meyniel graphs are perfect (Meyniel 1976).
The numbers of connected Meyniel graphs on 
, 2, ... vertices
are 1, 1, 2, 6, 19, 88, 459, 3060, 23989, ... (the first few of which are illustrated
above), and the corresponding numbers of not-necessarily connected
Meyniel graphs are 1, 2, 4, 11, 32, 130, 622, 3839, ... (OEIS A123434).
By definition, any graph with no odd graph cycle of length at least 5 is trivially Meyniel. In particular, every bipartite graph, equivalently every bicolorable graph, is Meyniel because it has no odd cycles. Classes of graphs that are Meyniel include empty graphs (trivially), complete graphs, and trees (trivially).
