On String Contact Representations in 3D

TL;DR

This paper investigates the complexity of string contact representations of graphs in 3D, establishing conditions under which graphs can be represented with strings having fewer bends, and providing bounds on their complexity.

Contribution

It introduces new results on the existence of low-bend string contact representations for specific classes of graphs, such as Hamiltonian triangle-free and bipartite graphs.

Findings

Hamiltonian triangle-free graphs admit B_3-string contact representations.

3-regular bipartite graphs admit B_2-string contact representations.

Hamiltonian bipartite graphs can be represented with all but one string as B_1-shapes.

Abstract

An axis-aligned string is a simple polygonal path, where each line segment is parallel to an axis in $\mathbb{R}^3$. Given a graph $G$, a string contact representation $\Psi$ of $G$ maps the vertices of $G$ to interior disjoint axis-aligned strings, where no three strings meet at a point, and two strings share a common point if and only if their corresponding vertices are adjacent in $G$. The complexity of $\Psi$ is the minimum integer $r$ such that every string in $\Psi$ is a $B_r$-string, i.e., a string with at most $r$ bends. While a result of Duncan et al. implies that every graph $G$ with maximum degree 4 has a string contact representation using $B_4$-strings, we examine constraints on $G$ that allow string contact representations with complexity 3, 2 or 1. We prove that if $G$ is Hamiltonian and triangle-free, then $G$ admits a contact representation where all the strings but one are $B_3$-strings. If $G$ is 3-regular and bipartite, then $G$ admits a contact representation with string complexity 2, and if we further restrict $G$ to be Hamiltonian, then $G$ has a contact representation, where all the strings but one are $B_1$-strings (i.e., $L$-shapes). Finally, we prove some complementary lower bounds on the complexity of string contact representations.