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|Title:||On semicomplete multipartite digraphs whose king sets are semicomplete digraphs|
Semicomplete multipartite digraphs
|Citation:||Tan, B.P. (2008-06-28). On semicomplete multipartite digraphs whose king sets are semicomplete digraphs. Discrete Mathematics 308 (12) : 2564-2570. ScholarBank@NUS Repository. https://doi.org/10.1016/j.disc.2007.06.002|
|Abstract:||Reid [Every vertex a king, Discrete Math. 38 (1982) 93-98] showed that a non-trivial tournament H is contained in a tournament whose 2-kings are exactly the vertices of H if and only if H contains no transmitter. Let T be a semicomplete multipartite digraph with no transmitters and let Kr (T) denote the set of r-kings of T. Let Q be the subdigraph of T induced by K4 (T). Very recently, Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] proved that Q contains no transmitters and gave an example to show that the direct extension of Reid's result to semicomplete multipartite digraphs with 2-kings replaced by 4-kings is not true. In this paper, we (1) characterize all semicomplete digraphs D which are contained in a semicomplete multipartite digraph whose 4-kings are exactly the vertices of D. While it is trivial that K4 (Q) ⊆ K4 (T), Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] showed that K3 (Q) ⊆ K3 (T) and K2 (Q) = K2 (T). Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] also provided an example to show that K3 (Q) need not be the same as K3 (T) in general and posed the problem: characterize all those semicomplete multipartite digraphs T such that K3 (Q) = K3 (T). In the course of proving our result (1), we (2) show that K3 (Q) = K3 (T) for all semicomplete multipartite digraphs T with no transmitters such that Q is a semicomplete digraph. © 2007 Elsevier B.V. All rights reserved.|
|Source Title:||Discrete Mathematics|
|Appears in Collections:||Staff Publications|
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