Arc-disjoint out-branchings and in-branchings in semicomplete digraphs

IF 0.9 3区 数学 Q2 MATHEMATICS
J. Bang-Jensen, Y. Wang
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引用次数: 0

Abstract

An out-branching B u + ${B}_{u}^{+}$ (in-branching B u ${B}_{u}^{-}$ ) in a digraph D $D$ is a connected spanning subdigraph of D $D$ in which every vertex except the vertex u $u$ , called the root, has in-degree (out-degree) one. It is well known that there exists a polynomial algorithm for deciding whether a given digraph has k $k$ arc-disjoint out-branchings with prescribed roots ( k $k$ is part of the input). In sharp contrast to this, it is already NP-complete to decide if a digraph has one out-branching which is arc-disjoint from some in-branching. A digraph is semicomplete if it has no pair of nonadjacent vertices. A tournament is a semicomplete digraph without directed cycles of length 2. In this paper we give a complete classification of semicomplete digraphs that have an out-branching B u + ${B}_{u}^{+}$ which is arc-disjoint from some in-branching B v ${B}_{v}^{-}$ where u , v $u,v$ are prescribed vertices of D $D$ . Our characterization, which is surprisingly simple, generalizes a complicated characterization for tournaments from 1991 by the first author and our proof implies the existence of a polynomial algorithm for checking whether a given semicomplete digraph has such a pair of branchings for prescribed vertices u , v $u,v$ and construct a solution if one exists. This confirms a conjecture of Bang-Jensen for the case of semicomplete digraphs.

半完整数图中的弧-不相连外分支和内分支
数图 D$D$ 中的外分支 Bu+${B}_{u}^{+}$(内分支 Bu-${B}_{u}^{-}$)是 D$D$ 的连跨子数图,其中除了顶点 u$u$(称为根)之外,每个顶点的内(外)度都是 1。众所周知,存在一种多项式算法来判定给定的图是否有 k$k$ 个带有规定根(k$k$ 是输入的一部分)的弧异节外分支。与此形成鲜明对比的是,判断一个数图是否有一个与某个内支弧相交的外支已经是 NP-complete。如果一个数图没有一对不相邻的顶点,那么它就是半完全数图。本文给出了半完整数图的完整分类,这些数图有一个外分支 Bu+${B}_{u}^{+}$,它与某个内分支 Bv-${B}_{v}^{-}$ 是弧相交的,其中 u,v$u,v$ 是 D$D$ 的规定顶点。我们的特征描述出奇地简单,它概括了第一作者 1991 年对锦标赛的复杂特征描述。我们的证明意味着存在一种多项式算法,可以检查给定的半完全数图是否有这样一对规定顶点 u,v$u,v$ 的分支,如果存在,则构建一个解。这证实了班-简森关于半完全图的猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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