Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to each vertex $u$ of $G$ such that $(u,v)\in E(G)$ if and only if $S_u\cap T_v\neq\emptyset$. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs.

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区间有向图中的核及相关问题

给定一个有向图 $G$，如果图中的每个顶点都在 $X$ 中或者是相邻的（resp），则称集合 $X\subseteq V(G)$ 是吸收集（resp.dominating set） $X$ 中顶点的外邻居）。如果$S$ 中没有两个顶点在$G$ 中相邻，则称集合$S\subseteq V(G)$ 是独立集。$G$ 的内核（相应的解决方案）是 $G$ 中的一个独立的和吸收的（相应的）集。我们在众所周知的区间有向图中探索这些问题的算法复杂性。一个有向图 $G$ 是一个区间有向图，如果一对区间 $(S_u,T_u)$ 可以分配给 $G$ 的每个顶点 $u$ 使得 $(u,v)\in E(G)$ 如果并且仅当 $S_u\cap T_v\neq\emptyset$ 时。通过限制可以分配给顶点的区间对的种类，文献中已经定义和研究了区间有向图的许多不同子类。我们观察到这些类中的几个，如区间捕获有向图、区间嵌套有向图、调整区间有向图和按时间顺序排列的区间有向图，是更一般的自反区间有向图类的子类——当我们要求将两个区间分配给一个顶点必须相交。我们证明了上面提到的所有问题都是有效可解的，在大多数情况下甚至是线性时间可解的，属于自反区间有向图的类别，但即使是非常有限的区间有向图称为点对点有向图也是 APX-hard ，其中分配给每个顶点的两个区间需要退化，即 它们各由一个点组成。我们获得的结果改进和推广了自反区间有向图的子类的几种现有算法和结构结果。