论文标题
挖掘中的强数
Strong in-domatic number in digraphs
论文作者
论文摘要
令$ d =（v，a）$为digraph，$ \ mathfrak {s} $ $ v（d）$的分区。我们说$ \ mathfrak {s} $是一个强大的分区分区，如果每个$ \ mathfrak中的每个$ s $ in Mathfrak {s} $都认为，每个顶点不在$ s $中至少具有$ s $中的一个外部neighbor，那就是$ s $，那就是$ s $ in neighbor in n op s opoys set，$ d \ d \ langle s \ langle s \ rangle $ rangle s stromp and comptern comptern comptern contect。强内部分区中的最大元素数量称为$ d $的强内部数字，它用$ \ mathsf {d} _ {s}^{ - }（d）$表示。在本文中，我们介绍了这些概念，并确定$ \ mathsf {d} _ {s}^{ - } $的$ \ mathsf {d} _ {s}^{ - } $，用于半完整的挖掘和平面挖掘。我们显示了具有强大的内部分区的挖掘物的一些结构属性，并且我们看到了$ \ Mathsf {d} _ {s}^{ - }（d）$的一些界限。然后，我们在笛卡尔产品，组成，线路挖掘和其他相关的挖掘物中研究了这个概念。 此外，我们表征了强大的内部关键挖掘物，并为两个家庭提供了具有某些属性的强大临界挖掘物，其中一个强大的临界差异$ d $认为$ \ mathsf {d} _ {s} _ {s}^{ - }^ - }（d-e）（d-e） $ a（d）$。
Let $D=(V,A)$ be a digraph and $\mathfrak{S}$ a partition of $V(D)$. We say that $\mathfrak{S}$ is a strong in-domatic partition if every $S$ in $\mathfrak{S}$ holds that every vertex not in $S$ has at least one out-neighbor in $S$, that is $S$ is an in-dominating set, and $D\langle S \rangle$ is strongly connected. The maximum number of elements in a strong in-domatic partition is called the strong in-domatic number of $D$ and it is denoted by $\mathsf{d}_{s}^{-}(D)$. In this paper we introduce those concepts and determine the value of $\mathsf{d}_{s}^{-}$ for semicomplete digraphs and planar digraphs. We show some structural properties of digraphs which have a strong in-domatic partition and we see some bounds for $\mathsf{d}_{s}^{-}(D)$. Then we study this concept in the Cartesian product, composition, line digraph and other associated digraphs. In addition, we characterize strong in-domatic critical digraphs and we give two families strong in-domatic critical digraphs which hold some properties, where a strong in-domatic critical digraph $D$ holds that $\mathsf{d}_{s}^{-}(D-e) = \mathsf{d}_{s}^{-}(D) -1 $ for every $e$ in $A(D)$.
