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.
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Research supported by N.S.E.R.C. grants #69-0982 and #69-0259.
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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
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DOI: https://doi.org/10.1007/BF01788692