Intersection Graphs of L-Shapes and Segments in the Plane
Stefan Felsner, Kolja Knauer, George B. Mertzios, Torsten Ueckerdt

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 -bend path is a simple path in the plane, whose direction changes 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 -bend path, or a segment, then this graph is called an -graph, -graph, -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365--372, 1992], stating that every -graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are -graphs, or -VPG-graphs for some small constant . We show that all planar -trees, all line graphs of planar graphs, and all full subdivisions of…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Structural Analysis and Optimization · Advanced Graph Theory Research
