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

TL;DR

This paper introduces a novel approach using ray-shooting queries to develop sublinear algorithms for finding minimum dominating sets in RDV graphs, improving efficiency over previous methods.

Contribution

It presents a new algorithm leveraging ray-shooting data structures to compute minimum dominating sets in RDV graphs in near-linear time, extending techniques to interval graphs.

Findings

Minimum dominating set in RDV graphs can be found in O(n log n) time.

Ray-shooting data structures enable sublinear algorithms for domination problems.

The approach offers a new proof for minimum dominating sets in interval graphs in O(n) time.

Abstract

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.