The excluded minors for isometric realizability in the plane

TL;DR

This paper identifies the complete set of excluded minors for graphs with isometric realizability in the plane under the maximum norm, revealing specific minimal obstructions and properties of the parameter across graph classes.

Contribution

It determines the exact excluded minors for the property $f_ abla(G) \,\leq 2$ for the maximum norm, and shows unboundedness of $f_ abla$ on planar graphs and in relation to tree-width.

Findings

Excluded minors for $f_ abla(G) \,\leq 2$ are the wheel on 5 vertices and a specific glued $K_4$ graph.

$f_ abla$ is unbounded on planar graphs.

$f_ abla$ is not bounded as a function of tree-width.

Abstract

Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $$\|r_v-r_w\|_p=\|q_v-q_w\|_p, \ \text{ for all }\ vw\in E(G).$$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property $f_p(G) \leq k$. In this paper, we determine the complete set of excluded minors for $f_\infty(G) \leq 2$. The two excluded minors are the wheel on $5$ vertices and the graph obtained by gluing two copies of $K_4$ along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for $f_1(G) \leq 2$. In addition, we give a family of examples that show that $f_\infty$ is unbounded on the class of planar graphs and $f_\infty$ is not bounded as a function of tree-width.