Intersection Graphs of L-Shapes and Segments in the Plane

Stefan B Felsner (1) , Kolja Knauer (2) , George B Mertzios (3) , Torsten Ueckerdt (4)

1 Discrete Mathematics [Berlin]
2 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
3 School of Engineering and Computing Sciences
4 KIT - Karlsruhe Institute of Technology

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, L , L and L. 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 L , a k-bend path, or a segment, then this graph is called an {L}-graph, {L, L }-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, L }-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 complements of pla-nar 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.

Mots clés: intersection graphs; segment graphs; co-planar graphs; k- bend VPG-graphs; planar 3-trees

Domaines: Informatique [cs]

Stefan B Felsner, Kolja Knauer, George B Mertzios, Torsten Ueckerdt. Intersection Graphs of L-Shapes and Segments in the Plane. Discrete Applied Mathematics, 2016, 206, pp.48-55. ⟨10.1016/j.dam.2016.01.028⟩. ⟨hal-01457800⟩