Abstract
A digraph is locally-in semicomplete if for every vertex of D its in-neighborhood induces a semicomplete digraph and it is locally semicomplete if for every vertex of D the in-neighborhood and the out-neighborhood induces a semicomplete digraph. The locally semicomplete digraphs where characterized in 1997 by Bang-Jensen et al. and in 1998 Bang-Jensen and Gutin posed the problem if finding a kernel in a locally-in semicomplete digraph is polynomial or not. A kernel of a digraph is a set of vertices, which is independent and absorbent. A digraph D such that every proper induced subdigraph of D has a kernel is said to be critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. A digraph without an induced CKI-digraph as a subdigraph does have a kernel. We characterize the locally semicomplete digraphs, which are CKI. As a consequence of this characterization we conclude that determinate whether a locally semicomplete digraph is a CKI-digraph or not, is polynomial.
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We thanks the anonymous referee for the comments and suggestions, which improved substantially the rewriting of this paper.
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Research supported by CONACYT 219840 and UNAM-DGAPA-PAPIIT IN106613.
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Galeana-Sánchez, H., Olsen, M. A Characterization of Locally Semicomplete CKI-digraphs. Graphs and Combinatorics 32, 1873–1879 (2016). https://doi.org/10.1007/s00373-016-1708-9
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DOI: https://doi.org/10.1007/s00373-016-1708-9