A sufficient condition for the existence of plane spanning trees on geometric graphs

TL;DR

This paper establishes a sufficient condition based on empty triangles for the existence of a non-self intersecting spanning tree in geometric graphs, advancing understanding of geometric graph connectivity.

Contribution

It introduces a new criterion involving empty triangles that guarantees the existence of a plane spanning tree in geometric graphs.

Findings

The condition ensures a spanning tree exists if the number of disconnected empty triangles is at most n-3.

The result applies to points in general position in the plane.

Provides a new geometric criterion for spanning tree existence.

Abstract

Let P be a set of n > 2 points in general position in the plane and let G be a geometric graph with vertex set P. If the number of empty triangles uvw in P for which the subgraph of G induced by {u,v,w} is not connected is at most n-3, then G contains a non-self intersecting spanning tree.