Intersection Graphs of L-Shapes and Segments in the Plane

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, L , Land 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, Bk-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 Bk-VPG-graphs for some small constant k. We show that all planar 3trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are {L}-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.