T-shape visibility representations of 1-planar graphs

Abstract 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 of width ϵ > 0 between the polygons assigned to its endvertices. Special shapes are rectangles, L , T , E , and H -shapes, and caterpillars. A graph is 1-planar if there is a drawing in the plane such that each edge is crossed at most once and is IC-planar if in addition no two crossed edges share a vertex. We show that every IC-planar graph has a flat rectangle visibility representation, in which each rectangle has height ϵ, and that every 1-planar graph has a T -shape visibility representation. The representations use quadratic area and can be computed in linear time from a given embedding.

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