On the orientation of meyniel graphs

A kernel of a directed graph is a set of vertices K that is both absorbant and independent (i.e., every vertex not in K is the origin of an arc whose extremity is in K, and no arc of the graph has both endpoints in K). In 1983, Meyniel conjectured that any perfect graph, directed in such a way that every circuit of length three uses two reversible arcs, must have a kernel. This conjecture was proved for parity graphs. In this paper, we extend that result and prove that Meyniel's conjecture holds for all graphs in which every odd cycle has two chords.

[1]  Hortensia Galeana-Sánchez,et al.  On kernels and semikernels of digraphs , 1984, Discret. Math..

[2]  Frédéric Maffray,et al.  Kernels in perfect line-graphs , 1992, J. Comb. Theory, Ser. B.

[3]  Mostafa Blidia,et al.  A parity digraph has a kernel , 1986, Comb..

[4]  Christophe Champetier.  Kernels in some orientations of comparability graphs , 1989, J. Comb. Theory, Ser. B.

[5]  Frédéric Maffray,et al.  On kernels in i-triangulated graphs , 1986, Discret. Math..

[6]  Henry Meyniel,et al.  On the perfect graph conjecture , 1976, Discret. Math..

[7]  P. Duchet,et al.  Graphes Noyau-Parfaits , 1980 .

[8]  Pierre Duchet,et al.  Parity graphs are kernel-M-solvable , 1987, J. Comb. Theory, Ser. B.