Bounding the order of a graph using its diameter and metric dimension: a study through tree decompositions and VC dimension

TL;DR

This paper establishes new upper bounds on the size of graphs based on diameter and metric dimension, using tree decompositions and VC dimension, with implications for various graph classes.

Contribution

It introduces improved bounds on graph order in terms of diameter and metric dimension for specific graph classes, including trees, outerplanar, and minor-free graphs.

Findings

Bound n=O(kd^2) for trees and outerplanar graphs.

Derived bounds for graphs with bounded treewidth and chordal graphs.

Established VC dimension-based bounds for minor-free and rankwidth-bounded graphs.

Abstract

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric dimension $k$. In general, the bound $n\leq d^k+k$ is known to hold. We prove a bound of the form $n=\mathcal{O}(kd^2)$ for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width $w$ and length $\ell$, we obtain a bound of the form $n=\mathcal{O}(kd^2(2\ell+1)^{3w+1})$. This implies in particular that $n=\mathcal{O}(kd^{\mathcal{O}(1)})$ for graphs of constant treewidth and $n=\mathcal{O}(f(k)d^2)$ for chordal graphs, where $f$ is a doubly-exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomass\'e) as a tool, we prove the bounds $n\leq (dk+1)^{t-1}+1$ for $K_t$-minor-free graphs, and $n\leq (dk+1)^{d(3\cdot 2^{r}+2)}+1$ for graphs of rankwidth at most $r$.