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Journal Articles SIAM Journal on Discrete Mathematics Year : 2018

## Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

Laurent Beaudou
Florent Foucaud
Arnaud Mary
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• PersonId : 169966
• IdHAL : arnaud-mary
Aline Parreau

#### 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 Thomassé) 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 $r$.

#### Domains

Combinatorics [math.CO]

### Dates and versions

hal-01848648 , version 1 (25-07-2018)

### Identifiers

• HAL Id : hal-01848648 , version 1
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### Cite

Laurent Beaudou, Peter Dankelmann, Florent Foucaud, Michael Antony Henning, Arnaud Mary, et al.. Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension. SIAM Journal on Discrete Mathematics, 2018, 32 (2), pp.902 - 918. ⟨10.1137/16M1097833⟩. ⟨hal-01848648⟩

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