Editing Perfect graph (section)

===Strong perfection===
The [[strongly perfect graph]]s are graphs in which, in every induced subgraph, there exists an independent set that intersects all [[maximal clique]]s. In the [[Meyniel graph]]s or ''very strongly perfect graphs'', every vertex belongs to such an independent set. The Meyniel graphs can also be characterized as the graphs in which every odd cycle of length five or more has at least two chords.{{r|hoang-1987}}

[[File:Cubic matchstick graph.svg|thumb|upright|A [[parity graph]] that is neither [[Distance-hereditary graph|distance-hereditary]] nor [[Bipartite graph|bipartite]]]]
A [[parity graph]] is defined by the property that between every two vertices, all [[induced path]]s have equal parity: either they are all even in length, or they are all odd in length. These include the distance-hereditary graphs, in which all induced paths between two vertices have the same length,{{r|cd-1999}} and bipartite graphs, for which all paths (not just induced paths) between any two vertices have equal parity. Parity graphs are Meyniel graphs, and therefore perfect: if a long odd cycle had only one chord, the two parts of the cycle between the endpoints of the chord would be induced paths of different parity. The prism over any parity graph (its [[Cartesian product of graphs|Cartesian product]] with a single edge) is another parity graph, and the parity graphs are the only graphs whose prisms are perfect.{{r|jansen-1998}}