Abstract
We prove that if a directed graph,D, contains no odd directed cycle and, for all but finitely many vertices, EITHER the in-degrees are finite OR the out-degrees are at most one, thenD contains an independent covering set (i.e. there is a kernel). We also give an example of a countable directed graph which has no directed cycle, each vertex has out-degree at most two, and which has no independent covering set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berge, C.: Graphes et hypergraphes. Paris: Dunod 1970
Duchet, P.: Graphes noyau-parfaits. In: Proc. of the Joint Franco-Canadian Coll. Ann. Discrete Math.9, 93–101 (1980)
Duchet, P., Meyniel, H., Une généralisation du théorème de Richardson sur l'existence de noyaux dans les graphes orientés. Discrete Math.43, 21–27 (1983)
Galeana-Sanchez, H., Neumann-Lara, V.: On kernels and semi-kernels of digraphs. Discrete Math.48, 67–76 (1984)
Richardson, M.: Solutions of irreflexive relations. Ann. of Math.58, 573–580 (1953)
Von Neumann, J., Morgenstern, O.: Theory of games and economic behaviour. Princeton, NJ: Princeton Univ. Press 1944
Author information
Authors and Affiliations
Additional information
Research supported by N.S.E.R.C. grants #69-0982 and #69-0259.
Rights and permissions
About this article
Cite this article
Milner, E.C., Woodrow, R.E. On directed graphs with an independent covering set. Graphs and Combinatorics 5, 363–369 (1989). https://doi.org/10.1007/BF01788692
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01788692