Intersections, circuits, and colorability of line segments

TL;DR

This paper establishes bounds on intersections and regions in structured line segment sets, explores their colorability related to the EFL Conjecture, and analyzes the complexity of related optimization problems.

Contribution

It provides new bounds for intersections and regions in specific graph-structured segment sets and investigates the complexity of segment coloring problems related to the EFL Conjecture.

Findings

Derived sharp bounds for intersections and regions in various graph-structured segments.

Analyzed the complexity of segment coloring problems related to the EFL Conjecture.

Proposed computational approaches for segment coloring optimization.

Abstract

We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets. We also apply these results to a variant of the Erd\H{o}s-Faber-Lov\'asz (EFL) Conjecture stating that the intersection points of $m$ segments can be colored with $m$ colors so that no segment contains points with the same color. We investigate an optimization problem related to the EFL Conjecture for line segments, determine its complexity, and provide some computational approaches.