Digraphs+Theory,+Algorithms+and+Applications_Part12 - 4.11...

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Unformatted text preview: 4.11 Locally Semicomplete Digraphs 203 digraphs first proved by Bang-Jensen, Guo, Gutin and Volkmann [55]. In the process of deriving this classification, we will show several important properties of locally semicomplete digraphs. We start our consideration from round digraphs, a nice special class of locally semicomplete digraphs. 4.11.1 Round Digraphs A digraph on n vertices is round if we can label its vertices v 1 ,v 2 ,...,v n so that for each i , we have N + ( v i ) = { v i +1 ,...,v i + d + ( v i ) } and N- ( v i ) = { v i- d- ( v i ) ,...,v i- 1 } (all subscripts are taken modulo n ). We will refer to the ordering v 1 ,v 2 ,...,v n as a round labelling of D . See Figure 4.15 for an example of a round digraph. Observe that every strong round digraph D is hamiltonian, since v 1 v 2 ...v n v 1 form a hamiltonian cycle, whenever v 1 ,v 2 ,...,v n is a round labelling. Round digraphs form a subclass of lo- cally semicomplete digraphs. We will see below that round digraphs play an important role in the study of locally semicomplete digraphs. 1 3 4 5 6 R 2 Figure 4.15 A round digraph with a round labelling. Proposition 4.11.1 [438] Every round digraph is locally semicomplete. Proof: Let D be a round digraph and let v 1 ,v 2 ,...,v n be a round labelling of D . Consider an arbitrary vertex, say v i . Let x,y be a pair of out-neighbours of v i . We show that x and y are adjacent. Assume without loss of generality that v i ,x,y appear in that circular order in the round labelling. Since v i → y and the in-neighbours of y appear consecutively preceding y , we must have x → y . Thus the out-neighbours of v i are pairwise adjacent. Similarly, we can show that the in-neighbours of v i are also pairwise adjacent. Therefore, D is locally semicomplete. ut In the rest of this subsection, we will prove the following characterization of round digraphs due to Huang [438]. This characterization generalizes the corresponding characterizations of round local tournaments and tournaments, due to Bang-Jensen [44] and Alspach and Tabib [22], respectively. 204 4. Classes of Digraphs ( d ) ( a ) ( b ) ( c ) Figure 4.16 Some forbidden digraphs in Huang’s characterization An arc xy of a digraph D is ordinary if yx is not in D . A cycle or path Q of a digraph D is ordinary if all arcs of Q are ordinary. To prove Theorem 4.11.4 below, we need two lemmas due to Huang [438]. Lemma 4.11.2 Let D be a round digraph then the following is true: (a) Every induced subdigraph of D is round. (b) None of the digraphs in Figure 4.16 is an induced subdigraph of D . (c) For each x ∈ V ( D ) , the subdigraphs induced by N + ( x )- N- ( x ) and N- ( x )- N + ( x ) are transitive tournaments....
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Digraphs+Theory,+Algorithms+and+Applications_Part12 - 4.11...

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