Intersection Graphs of L-Shapes and Segments in the Plane

Felsner, S.; Knauer, K.; Mertzios, G.B.; Ueckerdt, T.

Abstract

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: Full-size image (25 K), Full-size image (25 K),Full-size image (25 K) and Full-size image (25 K). 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 Full-size image (25 K), an Full-size image (25 K) or Full-size image (25 K), a k-bend path, or a segment, then this graph is called an {Full-size image (25 K)}-graph, {Full-size image (25 K),Full-size image (25 K)}-graph, Bk-VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer (1992), stating that every {Full-size image (25 K),Full-size image (25 K)}-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are {Full-size image (25 K)}-graphs, or Bk-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 {Full-size image (25 K)}-graphs. Furthermore we show that complements of planar graphs are B17-VPG-graphs and complements of full subdivisions are B2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.

Citation

Felsner, S., Knauer, K., Mertzios, G., & Ueckerdt, T. (2016). Intersection Graphs of L-Shapes and Segments in the Plane. Discrete Applied Mathematics, 206, 48-55. https://doi.org/10.1016/j.dam.2016.01.028

• Journal Article Type: Article
• Acceptance Date: Jan 22, 2016
• Online Publication Date: Feb 19, 2016
• Publication Date: Jun 1, 2016
• Deposit Date: Feb 12, 2016
• Publicly Available Date: Feb 19, 2017
• Journal: Discrete Applied Mathematics
• Print ISSN: 0166-218X
• Electronic ISSN: 1872-6771
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 206
• Pages: 48-55
• DOI: https://doi.org/10.1016/j.dam.2016.01.028
• Public URL: https://durham-repository.worktribe.com/output/1388579