Strong arc decompositions of split digraphs

IF 0.9 3区 数学 Q2 MATHEMATICS
Jørgen Bang-Jensen, Yun Wang
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引用次数: 0

Abstract

A strong arc decomposition of a digraph D = ( V , A ) is a partition of its arc set A into two sets A 1 , A 2 such that the digraph D i = ( V , A i ) is strong for i = 1 , 2 . Bang-Jensen and Yeo conjectured that there is some K such that every K -arc-strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3-arc-strong locally semicomplete digraph has a strong arc decomposition. A split digraph is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partitioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph D = ( V , A ) by adding a new set V of vertices and some arcs between V and V . In this paper, we prove that every 3-arc-strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2-strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs.

Abstract Image

分裂图的强弧分解
一个数图的强弧分解是将其弧集划分为两个集,使得该数图对 。Bang-Jensen 和 Yeo 猜想,存在这样一种情况,即每个-弧强的数图都有一个强弧分解。他们还证明,除了在四个顶点上有一个例外,每个 2 弧强半完全数图都有一个强弧分解。Bang-Jensen 和 Huang 将这一结果扩展到局部半完全数图,证明了每一个不是偶数循环的平方的 2 弧强局部半完全数图都具有强弧分解。这意味着每一个 3 弧强局部半完全数图都有一个强弧分解。分裂图是指底层无向图为分裂图的图,即其顶点可划分为一个小群和一个独立集。等价地,分裂数图是任何数图,它可以通过添加新的顶点集和一些与之间的弧从半完整数图中得到。 在本文中,我们证明了每一个 3 弧强分裂数图都有一个强弧分解,可以在多项式时间内找到,我们还提供了无穷类没有强弧分解的 2 强分裂数图。我们还提出了一些关于分裂图的未决问题。
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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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