Connectivity properties of locally semicomplete digraphs

Guo, Yubao; Volkmann, Lutz

doi:10.1002/jgt.3190180306

Cited by 26 publications

(15 citation statements)

References 6 publications

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“…Another kind of decomposition theorem for nonstrong locally semicomplete digraphs was described in [45]. Some properties of the strong components of a locally in-semicomplete digraph are described in the following two results from [27].…”

Section: Strong Componentsmentioning

confidence: 97%

Generalizations of tournaments: A survey

Bang‐Jensen¹,

Gutin

1998

J. Graph Theory

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We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these "tournament-like" digraphs.

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Section: Strong Componentsmentioning

confidence: 97%

Generalizations of tournaments: A survey

Bang‐Jensen¹,

Gutin

1998

J. Graph Theory

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“…Hence, if |A(D)|>n + 1, then D contains at least two non-critical vertices. This result was first obtained by Guo and Volkmann in [6] (see Theorem 2.2 therein) with quite a different method. Moreover, Meierling and Volkmann completely characterized in [9] the class of strong local tournaments with exactly two non-critical vertices.…”

Section: A1 Locally Semi-complete Digraphsmentioning

confidence: 67%

“…More precisely, we consider locally semi-complete and quasi-transitive digraphs (see Sections A.1 and A.2, respectively) and also multipartite tournaments which are orientations of complete multipartite (undirected) graphs (see Section A.3). On the basis of the theory of maximal proper strong subdigraphs in strong digraphs, we present a new result on non-critical vertices in strong quasi-transitive digraphs (see Corollary 5 in Section A.2) and give new proofs of the known results on non-critical vertices obtained in [1,3,6,15] by distinct authors (J. Bang-Jensen, J. Huang, Y. Guo, L. Volkmann, and M. Tewes) with the use of different methods. Hence, the goal of the Appendix is two-fold.…”

Section: Appendix A: Non-critical Vertices In Generalizations Of Tourmentioning

confidence: 99%

Non‐Critical Vertices and Long Circuits in Strong Tournaments of Order n and Diameter d

Savchenko¹

2012

Journal of Graph Theory

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Let T be a strong tournament of order n≥4 with diameter d≥2. A vertex w in T is non‐critical if the subtournament T−w is also strong. In the opposite case, it is a critical vertex. In the present article, we show that T has at most max{2,d−1} critical vertices. This fact and Moon's vertex‐pancyclic theorem imply that for d≥3, it contains at least n−d+1 circuits of length n−1. We describe the class of all strong tournaments of order n≥4 with diameter d≥3 for which this lower bound is achieved and show that for 2h≤n−d+1, the minimum number of circuits of length n−h in a tournament of this class is equal to ()n−d+1h. In turn, the minimum among all strong tournaments of order n≥3 with diameter 2 grows exponentially with respect to n for any given h≥0. Finally, for n>d≥3, we select a strong tournament Td,n of order n with diameter d and conjecture that for any strong tournament T of order n whose diameter does not exceed d, the number of circuits of length ℓ in T is not less than that in Td,n for each possible ℓ. Moreover, if these two numbers are equal to each other for some given ℓ=n−d+3,...,d, where d≥(n+3)/2, then T is isomorphic to either Td,n or its converse Td,n−. For d=n−1, this conjecture was proved by Las Vergnas. In the present article, we confirm it for the case d=n−2. In an Appendix, some problems concerning non‐critical vertices are considered for generalizations of tournaments.

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“…A kind of the decomposition of non-strong locally semicomplete digraphs described in [17] is the following.…”

Section: The Structure Of a Local Tournamentmentioning

confidence: 99%

The second neighbourhood for bipartite tournaments

Li¹,

Sheng²

2019

Discuss. Math. Graph Theory

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Sullivan stated the conjectures: (1) every oriented graph D has a ver-In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament D, we prove that D has at least two vertices satisfying (1) if D has no vertex of in-degree zero. And, for a local tournament D, we prove that either there exist two vertices satisfying (2) or there exists a vertex v satisfying d ++ (v) + d + (v) ≥ 2d − (v) + 2 if D has no vertex of in-degree zero.

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Connectivity properties of locally semicomplete digraphs

Abstract: It is shown that every k-connected locally semicomplete digraph D with minimum outdegree at least 2k and minimum indegree at least 2k -2 has at least rn = max(2, k} vertices XI, x2, . . . , x, such that Dx; is k-connected for i 1,2,. . . , m.

Cited by 26 publications

References 6 publications

Generalizations of tournaments: A survey

Generalizations of tournaments: A survey

Non‐Critical Vertices and Long Circuits in Strong Tournaments of Order n and Diameter d

The second neighbourhood for bipartite tournaments

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