Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust Algorithms for Finding Triangles and Computing the Girth in Unit Disk and Transmission Graphs

Katharina Klost, Wolfgang Mulzer

TL;DR

The paper develops robust algorithms for core graph problems in restricted geometric domains: triangle detection and girth in unit disk graphs, and directed triangle detection in transmission graphs. It leverages geometric-graph lemmas—such as a high-degree vertex implying a local triangle among seven neighbors in unit disk graphs and a similar bound on bidirected neighborhoods in transmission graphs—to design fast, domain-aware procedures. The main results include an $O(n)$-time robust algorithm for both triangle finding and girth in unit disk graphs, and an $O(n+m)$-time robust algorithm for directed triangles in transmission graphs. These findings show that, within restricted domains, substantial speedups are achievable and raise interesting questions about extending sublinear robust techniques to broader graph classes.

Abstract

We describe optimal robust algorithms for finding a triangle and the unweighted girth in a unit disk graph, as well as finding a triangle in a transmission graph.In the robust setting, the input is not given as a set of sites in the plane, but rather as an abstract graph. The input may or may not be realizable as a unit disk graph or a transmission graph. If the graph is realizable, the algorithm is guaranteed to give the correct answer. If not, the algorithm will either give a correct answer or correctly state that the input is not of the required type.

Robust Algorithms for Finding Triangles and Computing the Girth in Unit Disk and Transmission Graphs

TL;DR

The paper develops robust algorithms for core graph problems in restricted geometric domains: triangle detection and girth in unit disk graphs, and directed triangle detection in transmission graphs. It leverages geometric-graph lemmas—such as a high-degree vertex implying a local triangle among seven neighbors in unit disk graphs and a similar bound on bidirected neighborhoods in transmission graphs—to design fast, domain-aware procedures. The main results include an -time robust algorithm for both triangle finding and girth in unit disk graphs, and an -time robust algorithm for directed triangles in transmission graphs. These findings show that, within restricted domains, substantial speedups are achievable and raise interesting questions about extending sublinear robust techniques to broader graph classes.

Abstract

We describe optimal robust algorithms for finding a triangle and the unweighted girth in a unit disk graph, as well as finding a triangle in a transmission graph.In the robust setting, the input is not given as a set of sites in the plane, but rather as an abstract graph. The input may or may not be realizable as a unit disk graph or a transmission graph. If the graph is realizable, the algorithm is guaranteed to give the correct answer. If not, the algorithm will either give a correct answer or correctly state that the input is not of the required type.
Paper Structure (7 sections, 7 theorems)

This paper contains 7 sections, 7 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unit disk graphs
  4. Finding a Triangle
  5. Computing the Girth
  6. Finding a directed triangle in a transmission graph
  7. Conclusion

Key Result

Lemma 2

Let $G = (V, E)$ be a graph that is realizable as an unit disk graph, and let $v \in V$ be a vertex with $\deg(v) > 5$. Then, the subgraph induced by $v$ and any six adjacent vertices contains a triangle.

Theorems & Definitions (8)

  • Definition 1: Raghavan and Spinrad raghavan_robust_2003
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Theorem 9