Perfectly orderable graphs are quasi-parity graphs: a short proof

%'c shw that perfectly orderabk grapk Sa are q&-parity graphs by exhibiting two &lodes which are not llinked by a chordless odd chain. This proof is short and simpler than the one given by H. Meynid. A graph G = (V, E) _ is called a quasi-~&y graph if for every A s X the induced subgraph GA is a clique or contains 2 nodes x, y which are riot linked by a (chordless) odd chain in G,+ Quasi-parity graphs have been introduced by Meyniel[3]. A short proof that they are perfect graphs is given in [l]. V. Chvataf [2] has introduced the class of perfecdry orderable graphs. It consists of graphs for which an ordering < of the nodes can be found such that for any subgraph G' of 6, the sequential node coloring algorithm based on this induced ordering in G ' (" always use the smallest possible color ") gives an optimal coloring for G '. An ordering of the node:. of G has this property and is called perfect ordering iff it induces no obs~~ction. A chordless path with nodes a, b, c, d and edges [a, b], [b, c], [c, d] defines an obstruction on a, 6, c, d if we have a < 6 and d cc, We shall prove that perfectly orderable graphs are quasi-parity graphs, %leyuiel has given a proof of this result in [3]. is proof is quite different from ours; it is i-A *n ~3 induction process. Ours seems to be simpler. " ~ " YY VP. ._I_ We consider the family r " of all unordered pairs (x, z) of andes such hit x z are the endpoints of an induced P3 (chain ou nodes x, y, z). Given a ge ordering < of the nodes, we shall need the following definition: (x, 2) E I; is minimal if there is no pair (u, ip) + (x, 2) in F wit A pair of nonadjacent nodes a, 6 will be denoted [a].