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

Therese Biedl, Prashant Gokhale

TL;DR

This work addresses sublinear-time computation of a minimum dominating set in RDV graphs, represented as downward paths in rooted trees. The authors extend a segment-intersection framework with a ray-shooting data structure to implement two key operations in the classic greedy algorithm, achieving an overall time of $O(n \log n)$ given a linear-sized RDV representation. A separate, self-contained argument yields an $O(n)$-time algorithm for interval graphs by exploiting their RDV-path host-tree structure. The results demonstrate how geometric data structures can drive efficient domination algorithms in structured graph classes and hint at broader applicability to related problems. The approaches have potential practical impact in areas where RDV representations arise and edge counts are large but implicit.

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.

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

TL;DR

This work addresses sublinear-time computation of a minimum dominating set in RDV graphs, represented as downward paths in rooted trees. The authors extend a segment-intersection framework with a ray-shooting data structure to implement two key operations in the classic greedy algorithm, achieving an overall time of given a linear-sized RDV representation. A separate, self-contained argument yields an -time algorithm for interval graphs by exploiting their RDV-path host-tree structure. The results demonstrate how geometric data structures can drive efficient domination algorithms in structured graph classes and hint at broader applicability to related problems. The approaches have potential practical impact in areas where RDV representations arise and edge counts are large but implicit.

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 -vertex RDV graph, using priority search trees, in 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 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 time.
Paper Structure (10 sections, 7 theorems, 2 figures, 2 algorithms)

This paper contains 10 sections, 7 theorems, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. From RDV representation to segments
  4. Pre-processing the RDV representation
  5. Dominating Sets in RDV Graphs
  6. Implementing Operation $A$ in $O(\log{n})$ Time per Iteration
  7. Implementing Operation $B$ in $O(\log{n})$ Time per Iteration
  8. $O(n)$-time Algorithm for Interval Graphs
  9. Further Thoughts
  10. Missing Proof

Key Result

theorem 1

BG-CGT25 Let $G$ be a graph with an RDV representation and let $z_1,\dots,z_n$ be a bottom-up enumeration of vertices. Then for any $i<j$, edge $(z_i,z_j)$ exists if and only if the vertical segment $\mathbf{v}(z_j)$ intersects the horizontal segment $\mathbf{h}(z_i)$.

Figures (2)

  • Figure 1: An RDV graph with an RDV representation. Each node $\nu$ lists those vertices $z$ with $\nu\in T(z)$; we write $\underline{z}$ if $\nu=b(z)$ and $\overline{z}$ if $\nu=t(z)$. We also show the paths as poly-lines, with colors/dash-pattern matching the vertices. Nodes are drawn at their coordinates, and indices correspond to a bottom-up enumeration order.
  • Figure 2: The modified RDV representation, and the segments corresponding to vertices (drawn slightly offset for legibility).

Theorems & Definitions (13)

  • definition 1
  • definition 2
  • theorem 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • theorem 2
  • ...and 3 more