page 16 : boxicity – The boxicity of a graph $G$ is the minimum amount of interval graphs required, such that their intersection (ed: fixed typo) results in $G$.
page 26 : neighborhood diversity $k$ upper bounds boxicity by $k^{\mathcal O(1)}$ – Note that given a path of length $n > 3$. The boxicity is 1 while … neighborhood diversity is $n$. … a graph … with neighborhood diversity of $k$, its boxicity is at most $k+k^2$.
page 26 : bounded boxicity does not imply bounded neighborhood diversity – Note that given a path of length $n > 3$. The boxicity is 1 while … neighborhood diversity is $n$. … a graph … with neighborhood diversity of $k$, its boxicity is at most $k+k^2$.
page 35 : bounded c-closure does not imply bounded boxicity – Proposition 5.8. $c$-Closure is incomparable to Boxicity.
page 35 : bounded boxicity does not imply bounded c-closure – Proposition 5.8. $c$-Closure is incomparable to Boxicity.
page 42 : bounded degeneracy does not imply bounded boxicity – Proposition 7.1. Degeneracy is incomparable to Boxicity.
page 42 : bounded boxicity does not imply bounded degeneracy – Proposition 7.1. Degeneracy is incomparable to Boxicity.
page 8 : acyclic chromatic number $k$ upper bounds boxicity by $k^{\mathcal O(1)}$ – Lemma 4.9. The boxicity $b$ is upper bounded by the acyclic chromatic number $\chi_a$ (for every graph with $\chi_a>1$). We have $b \le \chi_a(\chi_a-1)$.
page 9 : boxicity $k$ upper bounds chordality by $\mathcal O(k)$ – Lemma 4.15 ([8,11]). The boxicity $b$ upper-bounds the chordality $c$. We have $c \le b$.
page 9 : distance to interval $k$ upper bounds boxicity by $\mathcal O(k)$ – Lemma 4.16. The distance $i$ to an interval graph upper bounds the boxicity $b$. We have $b \le i+1$.
page 9 : bounded distance to bipartite does not imply bounded boxicity – Theorem 2 For any $b \in \mathbb N^+$, there exists a chordal bipartite graph $G$ (ed: i.e. bipartite graph with no induced cycle on more than 4 vertices) with $\mathrm{box}(G) > b$.
