In this paper, we study the dominating set problem in \emph{RDV graphs}, a graph class that lies between interval graphs and chordal graphs and is defined as the \textbf{v}ertex-intersection graphs of \textbf{d}ownward paths in a \textbf{r}ooted tree. It was shown in a previous paper that adjacency queries in an RDV graph can be reduced to the question whether a horizontal segment intersects a vertical segment. This was then used to find a maximum matching in an $n$-vertex RDV graph, using priority search trees, in $O(n\log n)$ time, i.e., without even looking at all edges. In this paper, we show that if additionally we also use a ray shooting data structure, we can also find a minimum dominating set in an RDV graph $O(n\log n)$ time (presuming a linear-sized representation of the graph is given). The same idea can also be used for a new proof to find a minimum dominating set in an interval graph in $O(n)$ time.

翻译：本文研究\emph{RDV图}中的支配集问题。RDV图是介于区间图与弦图之间的图类，定义为有根树中\emph{向下路径的顶点交集图}。已有研究证明，RDV图中的邻接查询可转化为判断水平线段与垂直线段是否相交的问题，并借助优先搜索树在$O(n\log n)$时间内找到$n$顶点RDV图的最大匹配（无需遍历所有边）。本文进一步证明，若结合射线射击数据结构，可在给定图线性规模表示的前提下，以$O(n\log n)$时间求解RDV图的最小支配集。该思想还可为区间图最小支配集的$O(n)$时间算法提供新的证明框架。