Connected Components of Underlying Graphs of Halving Lines

TL;DR

This paper investigates the structure of underlying graphs formed by halving lines, demonstrating how to combine graphs and proving that each connected component is itself an underlying graph, advancing understanding of their combinatorial properties.

Contribution

It introduces methods to construct configurations with combined underlying graphs and proves that connected components are also underlying graphs, providing new insights into their structure.

Findings

Methods to combine underlying graphs of halving lines.

Proof that each connected component is an underlying graph.

New understanding of the structure of halving lines' underlying graphs.

Abstract

In this paper we discuss the connected components of underlying graphs of halving lines' configurations. We show how to create a configuration whose underlying graph is the union of two given underlying graphs. We also prove that every connected component of the underlying graph is itself an underlying graph.