T-shape visibility representations of 1-planar graphs

T-shape visibility representations of 1-planar graphs

A shape visibility representation displays a graph so that each vertex is represented by an orthogonal polygon of a particular shape and for each edge there is a horizontal or vertical line of sight between the polygons assigned to its endvertices. Special shapes are rectangles, L, T, E and H-shapes, and caterpillars. A flat rectangle is a horizontal bar of height > 0. A graph is 1-planar if th...

1-Visibility Representations of 1-Planar Graphs

A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertexsegments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices; beyond that segments do not intersect. We generalize visibility to 1-visibility, where each edge(vertex-) segment crosses at most one vertex(edge-) segment. ...

3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and...

On Visibility Representations of Non-Planar Graphs

A rectangle visibility representation (RVR) of a graph consists of an assignment of axisaligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding...

Rectangular and visibility representations of infinite planar graphs