具有反强连通性的混合弧不交生成子图的复杂性
The Complexity of Mixed Arc-Disjoint Spanning Subdigraphs with Antistrong Connectivity
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中文总结 AI 辅助
该研究证明了关于反强连通性的两个混合弧不交生成子图判定问题均为NP-完全，还给出了其在特定有向图类上的困难性结果，否定了多项式时间求解的可能性。
中文摘要 AI 辅助
若有向图D中的迹的弧交替为正向和反向，则该迹是反定向的。称有向图D是反强的，当且仅当对任意不同顶点对x,y∈V(D)，均存在一条正向反定向(x,y)-迹。Bang-Jensen、Bessy、Jackson和Kriesell[J. Combin. Theory Ser. B 122 (2017), 68--90]引入反强连通性，并提出两个关于混合弧不交生成子图的问题：其一需找到一个反强生成子图和一个弧不交的强生成子图；其二将强连通性替换为第二个子图的基础图为2-边连通的要求。Bang-Jensen等人询问这两个问题是否均可多项式时间求解，我们证明对应的两个判定问题均是NP-完全的。其中第一个问题在最大出度至多为4、最大入度至多为5的有向图上仍为NP-完全；第二个问题在强且反强、基础图为3-顶点连通、且至多两个顶点的入度和出度均大于4的定向有向图上仍为NP-完全，且该困难性结果不依赖于双弧（digons）。
英文摘要
A trail is antidirected if its arcs alternate between forward and backward. A digraph $D$ is antistrong if, for every ordered pair of distinct vertices $x,y\in V(D)$, it contains a forward antidirected $(x,y)$-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68--90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.
