On Computing Optimal Locally Gabriel Graphs

TL;DR

This paper introduces a new generalization of Locally Gabriel Graphs called Generalized Locally Gabriel Graphs, analyzes their computational complexity, and provides algorithms for verification.

Contribution

It defines GLGGs, proves NP-hardness and APX-hardness of related problems, and offers an algorithm to verify LGGs.

Findings

Computing maximum GLGG is NP-hard.

Computing LGG with dilation ≤ k is NP-hard.

Provided an algorithm to verify LGGs.

Abstract

Delaunay and Gabriel graphs are widely studied geometric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as \emph{Locally Delaunay Graphs} ($LDGs$) and \emph{Locally Gabriel Graphs} ($LGGs$) were proposed. We propose another generalization of $LGGs$ called \emph{Generalized Locally Gabriel Graphs} ($GLGGs$) in the context when certain edges are forbidden in the graph. Unlike a Gabriel Graph, there is no unique $LGG$ or $GLGG$ for a given point set because no edge is necessarily included or excluded. This property allows us to choose an $LGG/GLGG$ that optimizes a parameter of interest in the graph. We show that computing an edge maximum $GLGG$ for a given problem instance is NP-hard and also APX-hard. We also show that computing an $LGG$ on a given point set with dilation $\le k$ is NP-hard. Finally, we give an algorithm to verify whether a given geometric graph $G=(V,E)$ is a valid $LGG$.