Intersection Graphs of L-Shapes and Segments in the Plane

TL;DR

This paper explores the intersection representations of various subclasses of planar and subdivided graphs using L-shapes, segments, and k-bend paths, revealing new inclusions and bounds in geometric graph theory.

Contribution

It demonstrates that several important subclasses of planar and subdivided graphs can be represented as L-graphs or B_k-VPG-graphs for small k, extending known relationships in geometric intersection graph classes.

Findings

All planar 3-trees are L-graphs.

Line graphs of planar graphs are L-graphs.

Complements of planar graphs are B_{17}-VPG-graphs.

Abstract

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, \Gamma, LE{} and \eeG. A $k$-bend path is a simple path in the plane, whose direction changes $k$ times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an L, an L{} or \Gamma, a $k$-bend path, or a segment, then this graph is called an $\{L\}$-graph, $\{L,\Gamma\}$-graph, $B_k$-VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365--372, 1992], stating that every $\{L,\Gamma\}$-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are $\{L\}$-graphs, or $B_k$-VPG-graphs for some small constant $k$. We show that all planar $3$-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are $\{L\}$-graphs. Furthermore we show that all complements of planar graphs are $B_{17}$-VPG-graphs and all complements of full subdivisions are $B_2$-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.