Segment representation of a subclass of co-planar graphs

TL;DR

This paper proves that the complements of all partial 2-trees, a subclass of co-planar graphs, can be represented as segment graphs, advancing understanding of geometric representations of graph classes.

Contribution

It establishes that the complements of all partial 2-trees are segment graphs, answering a question about segment representations of certain co-planar graphs.

Findings

Complements of all partial 2-trees are segment graphs.

Addresses a question about segment representations of planar graph complements.

Expands the class of graphs known to have segment representations.

Abstract

A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochv\'{i}l and Kub\v{e}na [``On intersection representations of co-planar graphs'', Discrete Mathematics, 178(1-3):251-255, 1998] asked the question of whether the complements of planar graphs are segment graphs. We show here that the complements of all partial 2-trees are segment graphs.