AI 中文总结
本文研究涉及完美匹配、圈因子等性质的有向图弧分解问题，证明相关判定问题为NP完全，解决了该领域的若干开放问题。
AI 中文摘要
对于两个有向图性质$P_1$和$P_2$，有向图$D$的$(P_1,P_2)$-弧分解是指将弧集$A(D)$划分为$A_1\biguplus A_2$，使得生成子图$D[A_1]$和$D[A_2]$分别具有性质$P_1$和$P_2$。例如，有向图$D=(V,A)$的（强，强）-弧分解是将$A$划分为$A_1\bigcup A_2$，使得每个生成子图$D_i=(V,A_i)$（$i=1,2$）都是强连通的。本文证明，判定一个有向图是否允许具有$(P_1,P_2)\times$性质的弧分解是NP完全问题，其中$(P_1,P_2)$属于集合$\text{\textbraceleft}$（是完美匹配，无奇数有向圈）、（完美匹配，强连通）、（完美匹配，具有出分支）、（是圈因子，无奇数有向圈）$\text{\textbraceright}$。这些结果解决了Bang-Jensen、Bessy、Gonçalves和Picasarri-Arrieta提出的一些开放问题[《理论计算机科学》928卷（2022），第167--182页]。
英文摘要
For two digraph properties $P_1$ and $P_2$, a $(P_1,P_2)$-arc-decomposition of a digraph $D$ is a partition $A(D)=A_1\mathbin{\dot\cup}A_2$ such that the spanning subdigraphs $D[A_1]$ and $D[A_2]$ have properties $P_1$ and $P_2$, respectively. For example, a (strong,strong)-arc-decomposition of a digraph $D=(V,A)$ is a partitioning $A=A_1\cup{}A_2$ of $A$ so that each of the spanning digraphs $D_i=(V,A_i)$, $i=1,2$ are strongly connected. We prove that it is NP-complete to decide whether a digraph admits an arc-decomposition with properties $(P_1,P_2)$ where $(P_1,P_2)\in \{$(is a perfect matching, having no odd directed cycle), (perfect matching, strong), (perfect matching, having an out-branching), (is a cycle factor, having no odd directed cycle)$\}$. These results settle some open problems posed by Bang-Jensen, Bessy, Gonçalves, and Picasarri-Arrieta [Theoret. Comput. Sci. 928 (2022), 167--182].
Comments13 pages, 4 figures
