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

TL;DR

This paper presents optimal robust algorithms for identifying triangles and computing girth in unit disk and transmission graphs, handling inputs that may not be realizable as such graphs.

Contribution

It introduces algorithms that correctly identify graph properties or certify non-realizability in a robust manner, even with non-graph inputs.

Findings

Algorithms are optimal in the robust setting.

Correctly identify triangles and girth in realizable graphs.

Certify non-realizability when inputs are not valid graphs.

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.