Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounding the order of a graph using its diameter and metric dimension: a
  study through tree decompositions and VC dimension   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 $r$.

9
The categories of this paper are: nlin
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex