J. Bang-Jensen, Locally semicomplete digraphs: A generalizations of tournaments. J. Graph Theory 14 (1990) 371-390.  Home/Search   Document Not in Database   Summary   Related Articles   Check

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J. Bang-Jensen, Locally semicomplete digraphs: A generalizations of tournaments. J. Graph Theory 14 (1990) 371-390.

....strong demand of high degree for every pair of non adjacent vertices, by requiring this only for some pairs of non adjacent vertices. We do so by extending the following theorem on Hamiltonian locally semicomplete and out semicomplete digraphs (the locally semicomplete case was first proved in [2], the out semicomplete case was obtained in [6] Theorem 1.5 A strongly connected locally semicomplete (out semicomplete, respectively) digraph is Hamiltonian. Locally semicomplete digraphs include tournaments, and share many nice properties of tournaments (see, for example, 2, 3, 13] ....

....first proved in [2] the out semicomplete case was obtained in [6] Theorem 1.5 A strongly connected locally semicomplete (out semicomplete, respectively) digraph is Hamiltonian. Locally semicomplete digraphs include tournaments, and share many nice properties of tournaments (see, for example, [2, 3, 13]) Besides generalizing Theorem 1.5, our results, Theorems 4.1 and 4.2, generalize Theorems 1.1 and 1.2, respectively. In Section 2, we give necessary definitions and notation. Section 3 contains preliminary lemmas. We prove Theorems 4.1 and 4.2 in Section 4. These theorems neither imply nor are ....

J. Bang-Jensen, Locally semicomplete digraphs: a generalization of tournaments. J. Graph Theory 14 (1990) 371-390.

....a Hamiltonian path and every strongly connected tournament has a Hamiltonian cycle. It is an easy exercise to show that each of these results also hold for semicomplete digraphs a slight generalization of tournaments in which there is at least one arc between each pair of distinct vertices. In [2] the first author proved that the characterizations for Hamiltonian path and cycle in tournaments extend to locally semicomplete digraphs for every vertex x the set of inneighbours as well as the set of out neighbours of x induce a semicomplete digraph. He also showed that several other ....

....extend to locally semicomplete digraphs for every vertex x the set of inneighbours as well as the set of out neighbours of x induce a semicomplete digraph. He also showed that several other properties of tournaments hold for locally semicomplete digraphs as well. Since their introduction in [2], locally semicomplete digraphs have been extensively studied, see e.g. 2, 3, 7, 10, 11, 12, 14, 15, 16, 17, 18, 8, 19, 20, 23] Locally semicomplete digraphs are interesting, not just because they are a natural generalization of tournaments, but also because of their underlying undirected ....

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J. Bang-Jensen, Locally semicomplete digraphs: a generalization of tournaments, J. Graph Theory 14 (1990) 371-390.

....LSD s, for short) a This work was supported by the Danish Research Council under grant no. 11 0534 1. The support is gratefully acknowledged. 1 For definitions see the next section common generalization of two well studied families of digraphs, locally semicomplete digraphs (see e.g. [1, 2, 8, 16]) and extended semicomplete digraphs (see e.g. 4, 11, 12] It is shown that extended LSD s inherit some useful properties of both parents . Second, combining the results obtained for extended LSD s with some general results derived for totally Phi decomposable digraphs in [3] we prove the ....

....or replacing it with two oppositely oriented arcs. A locally semicomplete digraph (LSD, for short) is a digraph for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. It was proved [1] that every connected LSD has a Hamiltonian path. A digraph D is strongly connected (or just strong) if there exists an (x; y) Gammapath and a (y; x) Gammapath in D for every choice of distinct vertices x; y of D. It was shown [1] that every strong LSD has a Hamiltonian cycle. If a digraph is not ....

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J. Bang-Jensen, Locally semicomplete digraphs: a generalization of tournaments, J.Graph Theory 14 (1990) 371-390.

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J. Bang-Jensen, Locally semicomplete digraphs: A generalizations of tournaments. J. Graph Theory 14 (1990) 371-390.

....(other than the semicomplete digraphs ) to be studied. It was also known, see e.g. 54] that the hamiltonian cycle problem for semicomplete multipartite digraphs was quite difficult. A polynomial algorithm for this problem was recently found [21] This algorithm is based on results from [84] In [6] the class of locally semicomplete digraphs was introduced and it was proved that a number of structural properties of tournaments are still valid for this much larger class of digraphs. In particular the hamiltonian path and cycle problems are both easy for locally semicomplete digraphs : every ....

....component digraphs of generalizations of tournaments are not so simple but we still have some nice and useful properties here. The following theorem on strong components in a non strong extended locally semicomplete digraph generalizes the corresponding result for locally semicomplete digraphs [6]. Proposition 4.1 [14] Let D be a connected non strongly connected extended locally semicomplete digraph. 1. If A and B are distinct strong components of D, then either A)B, or B)A, or there are no arcs between A and B. 2. There is a unique ordering D 1 ; D s , s 2 of the strong ....

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J. Bang-Jensen, Locally semicomplete digraphs: a generalization of tournaments, J. Graph Theory 14 (1990) 371-390.

....None of the theorems 1.3, 1.5, 1.6, 1.7 imply each other. The conditions in Theorems 1.5 and 1. 6 where inspired by the fact that if D has no pair of non adjacent vertices with a common in neighbour or a common out neighbour, then D is a locally semicomplete digraph and it was shown in [1] that every strongly connected locally semicomplete digraph is Hamiltonian. In [2] the following generalization of Theorem 1.3 was proposed Conjecture 1.8 [2] Let D be a strong digraph. Suppose that d(x) d(y) 2n Gamma 1 for every pair of non adjacent vertices fx; yg with a common ....

J. Bang-Jensen, Locally semicomplete digraphs: a generalization of tournaments. J. Graph Theory 14 (1990) 371-390.

....are joined by an arc. A tournament is transitive if and only if it has no directed cycles. An oriented graph D = V; A) is a local tournament (local transitive tournament) if the set of out neighbours as well as the of in neighbours of each vertex v 2 V form a tournament (a transitive tournament) [2, 17]. The following lemma can be found in [17] see also [25] Lemma 4.3 Let G be an undirected graph then each of the following are equivalent. 1. G is a proper circular arc graph. 2. G is can be oriented as a local tournament digraph. 3. G can be oriented as a local transitive tournament digraph. ....

J. Bang - Jensen, Locally semicomplete digraphs-A generalization of tournaments. J. Graph Theory 14 (1990) 371 - 390.

....a hamiltonian path and every strongly connected tournament has a hamiltonian cycle. It is an easy exercise to show that each of these results also hold for semicomplete digraphs a slight generalization of tournaments in which there is at least one arc between each pair of distinct vertices. In [2] the first author proved that the characterizations for hamiltonian path and cycle in tournaments extend to locally semicomplete digraphs for every vertex x the set of inneighbours as well as the set of out neighbours of x induce a semicomplete digraph. He also showed that several other ....

....Dept. of Math. and Compt. Sci. Odense University, DK 5230, Odense, Denmark. This work was supported by the Danish Research Council under grant no. 11 0534 1. The support is gratefully acknowledged. x Lehrstuhl II fur Mathematik, RWTH Aachen, 52056 Aachen, Germany. Since their introduction in [2], locally semicomplete digraphs have been extensively studied, see e.g. 2, 3, 7, 5, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 21] Locally semicomplete digraphs are interesting, not just because they are a natural generalization of tournaments, but also because of their underlying undirected graphs. ....

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J. Bang-Jensen, Locally semicomplete digraphs: a generalization of tournaments, J. Graph Theory 14 (1990) 371-390.

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J. Bang-Jensen, Locally semicomplete digraphs: A generalizations of tournaments. J. Graph Theory 14 (1990) 371-390.

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J. Bang - Jensen, Locally semicomplete digraphs-A generalization of tournaments. J. Graph Theory 14 (1990) 371 - 390.

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