# T-Shape Visibility Representations of 1-Planar Graphs

## TL;DR

This paper introduces new shape visibility representations for 1-planar and IC-planar graphs, demonstrating that such graphs can be efficiently represented with flat rectangles and T-shapes in quadratic area.

## Contribution

It proves that IC-planar graphs admit flat rectangle visibility representations and 1-planar graphs admit T-shape visibility representations, both computable in linear time.

## Key findings

- IC-planar graphs have flat rectangle visibility representations.
- 1-planar graphs have T-shape visibility representations.
- Representations use quadratic area and are computable in linear time.

## 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 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 $\epsilon>0$. 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 crossing edges share a vertex. We show that every IC-planar graph has a flat rectangle visibility representation 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.