Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

TL;DR

This paper extends triangular-distance Delaunay graphs to higher orders, analyzing their connectivity, Hamiltonicity, perfect matchings, and edge blocking properties in the plane.

Contribution

It introduces and studies higher-order $k$-TD graphs, providing new insights into their structural properties and edge blocking challenges.

Findings

Higher-order $k$-TD graphs have specific connectivity thresholds.

Conditions for Hamiltonicity and perfect matchings are established.

Strategies for blocking edges in $k$-TD graphs are proposed.

Abstract

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an edge between two points in $P$ if and only if there is an empty homothet of $\triangledown$ having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains an edge between two points if the interior of the homothet of $\triangledown$ having the two points on its boundary contains at most $k$ points of $P$. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of $k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.