A Note on the Area Requirement of Euclidean Greedy Embeddings of Christmas Cactus Graphs

TL;DR

This paper proves that exponential area is necessary for Euclidean greedy embeddings of Christmas cactus graphs, resolving an open question about their minimal area requirements.

Contribution

It establishes that exponential size is unavoidable for greedy embeddings of Christmas cactus graphs, confirming the worst-case area complexity.

Findings

Greedy embeddings of Christmas cactus graphs require exponential area in the worst case.

The result confirms that exponential size is necessary for such embeddings.

This finding impacts the understanding of geometric routing in wireless networks.

Abstract

An Euclidean greedy embedding of a graph is a straight-line embedding in the plane, such that for every pair of vertices $s$ and $t$, the vertex $s$ has a neighbor $v$ with smaller distance to $t$ than $s$. This drawing style is motivated by greedy geometric routing in wireless sensor networks. A Christmas cactus is a connected graph in which every two simple cycles have at most one vertex in common and in which every cutvertex is part of at most two biconnected blocks. It has been proved that Christmas cactus graphs have an Euclidean greedy embedding. This fact has played a crucial role in proving that every 3-connected planar graph has an Euclidean greedy embedding. The proofs construct greedy embeddings of Christmas cactuses of exponential size, and it has been an open question whether exponential area is necessary in the worst case for greedy embeddings of Christmas cactuses. We prove that this is indeed the case.