Graphes Noyau-Parfaits

A directed graph G is said to be kernel-perfect if every G-subgraph possesses a kernel (= a stable dominating subset). Graphs without circuits, graphs without odd circuits are known to be kernel-perfect (Von Neumann and Morgenstern, and Richardson), and so are symmetric or transitive graphs. I present here the current state of research in kernel theory; the most significant results are the proofs of kernel-perfectness of G in each of the following cases: Theorem 2.2. G is a right- (or left-)pretransitive digraph. Theorem 3.3. (see [5]). Every odd circuit of G (not elementary in general) v 1 …v 2p+1 v 1 , possesses two crossing short chords (that means there exist two arcs of G of the form (v v u q+2 ) and (v q+1 , v q+3 ). Theorem 4.2. Every circuit of G possesses at least one symmetrical arc. Theorem 4.3. Every odd circuit of G possesses at least two symmetrical arcs. Theorems 3.3 and 4.3 are particuliar cases of an interesting conjecture proposed by H. Meyniel. Somes curious kernel-critical graphs are exhibited.