4-connected planar graphs are in $B_3$-EPG

TL;DR

This paper proves that all 4-connected planar graphs can be represented as $B_3$-EPG graphs, where each vertex is a curve with at most three bends sharing grid edges for adjacency.

Contribution

It introduces a new representation for 4-connected planar graphs as $B_3$-EPG graphs, extending previous models with a modified touching of thickened L-shapes.

Findings

All 4-connected planar graphs have a $B_3$-EPG representation.

The construction modifies existing L-shape representations.

The approach ensures adjacency corresponds to shared grid edges.

Abstract

We show that every 4-connected planar graph has a $B_3$-EPG representation, i.e., every vertex is represented by a curve on the grid with at most three bends, and two vertices are adjacent if and only if the corresponding curves share an edge of the grid. Our construction is based on a modification of the representation by touching thickened $L$-shapes proposed by Gon\c{c}alves et al.