Solving problems in RDV Graphs faster and easier

Overview & Synopsis

In interval graphs and many generalizations thereof, problems such as Matching, Independent Set, and Dominating Set can be solved with simple greedy algorithms in linear (i.e., $O(m+n)$) time. However, it is sometimes possible to improve on this runtime if a suitable representation as intersection graph is given; for example it is possible to find a minimum dominating set in an interval graph in $O(n)$ time. We study RDV graphs, a generalization of interval graphs, namely, the intersection graphs of directed paths in a rooted tree. It was known that the above three problems can be solved in $O(n\log n)$ time if a suitably small representation is given. In this paper, we reduce the runtime to $O(n)$. Furthermore, while the previous results used non-trivial operations such as range-minimum queries and ray-shooting queries, we need nothing more complicated than lists and (for Matching) a Union-Find data structure.