DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 24(2) (2004) 171-182
DOI: 10.7151/dmgt.1223

SOME SUFFICIENT CONDITIONS ON ODD DIRECTED CYCLES OF BOUNDED LENGTH FOR THE EXISTENCE OF A KERNEL

Hortensia Galeana-Sánchez

Instituto de Matemáticas, UNAM
Circuito Exterior, Ciudad Universitaria
04510 México, D.F. MEXICO

e-mail: hgaleana@matem.unam.mx

Abstract

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel-perfect digraph. In this paper I investigate some sufficient conditions for a digraph to have a kernel by asking for the existence of certain diagonals or symmetrical arcs in each odd directed cycle whose length is at most 2α(D)+1, where α(D) is the maximum cardinality of an independent vertex set of D. Previous results are generalized.

Keywords: kernel, kernel-perfect, critical kernel-imperfect.

2000 Mathematics Subject Classification: 05C20.

References

[1] J. Bang-Jensen, J. Huang and E. Prisner, In-Tournament Digraphs, J. Combin. Theory (B) 59 (1993) 267-287, doi: 10.1006/jctb.1993.1069.
[2] C. Berge, Graphs, North-Holland Mathematical Library, Vol. 6 (North-Holland, Amsterdam, 1985).
[3] C. Berge, Nouvelles extensions du noyau d'un graphe et ses applications en théorie des jeux, Publ. Econométriques 6 (1977).
[4] P. Duchet, Graphes Noyau-Parfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4.
[5] P. Duchet, A sufficient condition for a digraph to be kernel-perfect, J. Graph Theory 11 (1987) 81-85, doi: 10.1002/jgt.3190110112.
[6] P. Duchet and H. Meyniel, A note on kernel-critical graphs, Discrete Math. 33 (1981) 103-105, doi: 10.1016/0012-365X(81)90264-8.
[7] H. Galeana-Sánchez, Normal fraternally orientable graphs satisfy the strong perfect graph conjecture, Discrete Math. 122 (1993) 167-177, doi: 10.1016/0012-365X(93)90293-3.
[8] H. Galeana-Sánchez and V. Neumann-Lara, On kernels and semikernels of digraphs, Discrete Math. 48 (1984) 67-76, doi: 10.1016/0012-365X(84)90131-6.
[9] H. Galeana-Sánchez, B1 and B2-orientable graphs in kernel theory, Discrete Math. 143 (1995) 269-274, doi: 10.1016/0012-365X(94)00021-A.
[10] F. Gavril, V. Toledano and D. de Werra, Chordless paths, odd holes and kernels in graphs without M-obstructions, preprint.
[11] F. Gavril and J. Urrutia, An Algorithm for fraternal orientation of graphs, Information Processing Letters 41 (1993) 271-279.
[12] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1944).
[13] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573, doi: 10.2307/1969755.
[14] M. Richardson, Extensions theorems for solutions of irreflexive relations, Proc. Nat. Acad. Sci. USA. 39 (1953) 649, doi: 10.1073/pnas.39.7.649.
[15] D.J. Rose, Triangulated graphs and the elimination process, J. Math. Anal. Appl. 32 (1970) 597-609, doi: 10.1016/0022-247X(70)90282-9.
[16] D.J. Skrien, A relationship between triangulated graphs, comparability graphs, proper interval graphs, proper circular arc graphs, and nested interval graphs, J. Graph Theory 6 (1982) 309-316, doi: 10.1002/jgt.3190060307.

Received 6 May 2002
Revised 21 January 2004