Using Ray-shooting Queries for Sublinear Algorithms for Dominating Sets in RDV Graphs

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 $\mathbf{\text{v}}$ertex-intersection graphs of $\mathbf{\text{d}}$ownward paths in a $\mathbf{\text{r}}$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.